Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Percentage Accurate: 98.4% → 98.4%

Time: 17.5s

Alternatives: 27

Speedup: 1.0×

• 47.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * exp((((y * log(z)) + ((t - 1.0d0) * log(a))) - b))) / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.exp((((y * Math.log(z)) + ((t - 1.0) * Math.log(a))) - b))) / y;
}

Python

def code(x, y, z, t, a, b):
	return (x * math.exp((((y * math.log(z)) + ((t - 1.0) * math.log(a))) - b))) / y

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Initial Program: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * exp((((y * log(z)) + ((t - 1.0d0) * log(a))) - b))) / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.exp((((y * Math.log(z)) + ((t - 1.0) * Math.log(a))) - b))) / y;
}

Python

def code(x, y, z, t, a, b):
	return (x * math.exp((((y * math.log(z)) + ((t - 1.0) * math.log(a))) - b))) / y

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Alternative 1: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (+ t -1.0) (log a))) b))) y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * exp((((y * log(z)) + ((t + (-1.0d0)) * log(a))) - b))) / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.exp((((y * Math.log(z)) + ((t + -1.0) * Math.log(a))) - b))) / y;
}

Python

def code(x, y, z, t, a, b):
	return (x * math.exp((((y * math.log(z)) + ((t + -1.0) * math.log(a))) - b))) / y

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t + -1.0) * log(a))) - b))) / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
2. Add Preprocessing

3. Final simplification 99.2%

   \[\leadsto \frac{x \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \]
4. Add Preprocessing

Alternative 2: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := \frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -152:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a)))
        (t_2 (/ (* x (exp (- (* t (log a)) b))) y)))
   (if (<= t_1 -4e+16)
     t_2
     (if (<= t_1 -152.0)
       (/ x (* a (* y (exp b))))
       (if (<= t_1 2e+99) (/ (* x (exp (- (* y (log z)) b))) y) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double t_2 = (x * exp(((t * log(a)) - b))) / y;
	double tmp;
	if (t_1 <= -4e+16) {
		tmp = t_2;
	} else if (t_1 <= -152.0) {
		tmp = x / (a * (y * exp(b)));
	} else if (t_1 <= 2e+99) {
		tmp = (x * exp(((y * log(z)) - b))) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t + (-1.0d0)) * log(a)
    t_2 = (x * exp(((t * log(a)) - b))) / y
    if (t_1 <= (-4d+16)) then
        tmp = t_2
    else if (t_1 <= (-152.0d0)) then
        tmp = x / (a * (y * exp(b)))
    else if (t_1 <= 2d+99) then
        tmp = (x * exp(((y * log(z)) - b))) / y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * Math.log(a);
	double t_2 = (x * Math.exp(((t * Math.log(a)) - b))) / y;
	double tmp;
	if (t_1 <= -4e+16) {
		tmp = t_2;
	} else if (t_1 <= -152.0) {
		tmp = x / (a * (y * Math.exp(b)));
	} else if (t_1 <= 2e+99) {
		tmp = (x * Math.exp(((y * Math.log(z)) - b))) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t + -1.0) * math.log(a)
	t_2 = (x * math.exp(((t * math.log(a)) - b))) / y
	tmp = 0
	if t_1 <= -4e+16:
		tmp = t_2
	elif t_1 <= -152.0:
		tmp = x / (a * (y * math.exp(b)))
	elif t_1 <= 2e+99:
		tmp = (x * math.exp(((y * math.log(z)) - b))) / y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + -1.0) * log(a))
	t_2 = Float64(Float64(x * exp(Float64(Float64(t * log(a)) - b))) / y)
	tmp = 0.0
	if (t_1 <= -4e+16)
		tmp = t_2;
	elseif (t_1 <= -152.0)
		tmp = Float64(x / Float64(a * Float64(y * exp(b))));
	elseif (t_1 <= 2e+99)
		tmp = Float64(Float64(x * exp(Float64(Float64(y * log(z)) - b))) / y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + -1.0) * log(a);
	t_2 = (x * exp(((t * log(a)) - b))) / y;
	tmp = 0.0;
	if (t_1 <= -4e+16)
		tmp = t_2;
	elseif (t_1 <= -152.0)
		tmp = x / (a * (y * exp(b)));
	elseif (t_1 <= 2e+99)
		tmp = (x * exp(((y * log(z)) - b))) / y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Exp[N[(N[(t * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+16], t$95$2, If[LessEqual[t$95$1, -152.0], N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+99], N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -4e16 or 1.9999999999999999e99 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 94.3

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 94.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   if -4e16 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -152

   1. Initial program 96.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 84.2

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 84.2%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 90.5

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   if -152 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 1.9999999999999999e99

   1. Initial program 99.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 90.5

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 90.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -4 \cdot 10^{+16}:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -152:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := x \cdot \frac{{a}^{t}}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+105}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -152:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))) (t_2 (* x (/ (pow a t) y))))
   (if (<= t_1 -1e+105)
     t_2
     (if (<= t_1 -152.0)
       (/ x (* a (* y (exp b))))
       (if (<= t_1 5e+100) (/ (* x (exp (- (* y (log z)) b))) y) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double t_2 = x * (pow(a, t) / y);
	double tmp;
	if (t_1 <= -1e+105) {
		tmp = t_2;
	} else if (t_1 <= -152.0) {
		tmp = x / (a * (y * exp(b)));
	} else if (t_1 <= 5e+100) {
		tmp = (x * exp(((y * log(z)) - b))) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t + (-1.0d0)) * log(a)
    t_2 = x * ((a ** t) / y)
    if (t_1 <= (-1d+105)) then
        tmp = t_2
    else if (t_1 <= (-152.0d0)) then
        tmp = x / (a * (y * exp(b)))
    else if (t_1 <= 5d+100) then
        tmp = (x * exp(((y * log(z)) - b))) / y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * Math.log(a);
	double t_2 = x * (Math.pow(a, t) / y);
	double tmp;
	if (t_1 <= -1e+105) {
		tmp = t_2;
	} else if (t_1 <= -152.0) {
		tmp = x / (a * (y * Math.exp(b)));
	} else if (t_1 <= 5e+100) {
		tmp = (x * Math.exp(((y * Math.log(z)) - b))) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t + -1.0) * math.log(a)
	t_2 = x * (math.pow(a, t) / y)
	tmp = 0
	if t_1 <= -1e+105:
		tmp = t_2
	elif t_1 <= -152.0:
		tmp = x / (a * (y * math.exp(b)))
	elif t_1 <= 5e+100:
		tmp = (x * math.exp(((y * math.log(z)) - b))) / y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + -1.0) * log(a))
	t_2 = Float64(x * Float64((a ^ t) / y))
	tmp = 0.0
	if (t_1 <= -1e+105)
		tmp = t_2;
	elseif (t_1 <= -152.0)
		tmp = Float64(x / Float64(a * Float64(y * exp(b))));
	elseif (t_1 <= 5e+100)
		tmp = Float64(Float64(x * exp(Float64(Float64(y * log(z)) - b))) / y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + -1.0) * log(a);
	t_2 = x * ((a ^ t) / y);
	tmp = 0.0;
	if (t_1 <= -1e+105)
		tmp = t_2;
	elseif (t_1 <= -152.0)
		tmp = x / (a * (y * exp(b)));
	elseif (t_1 <= 5e+100)
		tmp = (x * exp(((y * log(z)) - b))) / y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[Power[a, t], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+105], t$95$2, If[LessEqual[t$95$1, -152.0], N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+100], N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -9.9999999999999994e104 or 4.9999999999999999e100 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 94.0

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 94.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {a}^{t}}{y}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{{a}^{t}}{y}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{{a}^{t}}{y}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{{a}^{t}}{y}} \]

      4. pow-lowering-pow.f64 91.6

         \[\leadsto x \cdot \frac{\color{blue}{{a}^{t}}}{y} \]

   8. Simplified 91.6%

      \[\leadsto \color{blue}{x \cdot \frac{{a}^{t}}{y}} \]

   if -9.9999999999999994e104 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -152

   1. Initial program 97.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 77.5

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 77.5%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 82.3

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 82.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   if -152 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 4.9999999999999999e100

   1. Initial program 99.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 89.8

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 89.8%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -1 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \frac{{a}^{t}}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -152:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{a}^{t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ t_2 := \frac{x \cdot {z}^{y}}{y}\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+39}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-177}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-300}:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (pow a (+ t -1.0))) y)) (t_2 (/ (* x (pow z y)) y)))
   (if (<= y -3.8e+39)
     t_2
     (if (<= y -1.95e-177)
       t_1
       (if (<= y -1.05e-300)
         (/ x (* y (exp b)))
         (if (<= y 4.6e+43) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * pow(a, (t + -1.0))) / y;
	double t_2 = (x * pow(z, y)) / y;
	double tmp;
	if (y <= -3.8e+39) {
		tmp = t_2;
	} else if (y <= -1.95e-177) {
		tmp = t_1;
	} else if (y <= -1.05e-300) {
		tmp = x / (y * exp(b));
	} else if (y <= 4.6e+43) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * (a ** (t + (-1.0d0)))) / y
    t_2 = (x * (z ** y)) / y
    if (y <= (-3.8d+39)) then
        tmp = t_2
    else if (y <= (-1.95d-177)) then
        tmp = t_1
    else if (y <= (-1.05d-300)) then
        tmp = x / (y * exp(b))
    else if (y <= 4.6d+43) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * Math.pow(a, (t + -1.0))) / y;
	double t_2 = (x * Math.pow(z, y)) / y;
	double tmp;
	if (y <= -3.8e+39) {
		tmp = t_2;
	} else if (y <= -1.95e-177) {
		tmp = t_1;
	} else if (y <= -1.05e-300) {
		tmp = x / (y * Math.exp(b));
	} else if (y <= 4.6e+43) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.pow(a, (t + -1.0))) / y
	t_2 = (x * math.pow(z, y)) / y
	tmp = 0
	if y <= -3.8e+39:
		tmp = t_2
	elif y <= -1.95e-177:
		tmp = t_1
	elif y <= -1.05e-300:
		tmp = x / (y * math.exp(b))
	elif y <= 4.6e+43:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y)
	t_2 = Float64(Float64(x * (z ^ y)) / y)
	tmp = 0.0
	if (y <= -3.8e+39)
		tmp = t_2;
	elseif (y <= -1.95e-177)
		tmp = t_1;
	elseif (y <= -1.05e-300)
		tmp = Float64(x / Float64(y * exp(b)));
	elseif (y <= 4.6e+43)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * (a ^ (t + -1.0))) / y;
	t_2 = (x * (z ^ y)) / y;
	tmp = 0.0;
	if (y <= -3.8e+39)
		tmp = t_2;
	elseif (y <= -1.95e-177)
		tmp = t_1;
	elseif (y <= -1.05e-300)
		tmp = x / (y * exp(b));
	elseif (y <= 4.6e+43)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -3.8e+39], t$95$2, If[LessEqual[y, -1.95e-177], t$95$1, If[LessEqual[y, -1.05e-300], N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+43], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.7999999999999998e39 or 4.6000000000000005e43 < y

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 96.2

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 96.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot {z}^{y}}}{y} \]

      3. pow-lowering-pow.f64 88.6

         \[\leadsto \frac{x \cdot \color{blue}{{z}^{y}}}{y} \]

   8. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

   if -3.7999999999999998e39 < y < -1.95000000000000007e-177 or -1.05000000000000002e-300 < y < 4.6000000000000005e43

   1. Initial program 98.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 82.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 82.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Simplified 73.2%

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y}} \]

   if -1.95000000000000007e-177 < y < -1.05000000000000002e-300

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 100.0

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 100.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 89.1

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 89.1%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]
   9. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{e^{b}}}}{y} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{e^{b}}}}{y} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      6. exp-lowering-exp.f64 89.1

         \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 89.1%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-177}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-300}:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {a}^{\left(t + -1\right)} \cdot \frac{x}{y}\\ t_2 := \frac{x \cdot {z}^{y}}{y}\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+40}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-204}:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (pow a (+ t -1.0)) (/ x y))) (t_2 (/ (* x (pow z y)) y)))
   (if (<= y -3.3e+40)
     t_2
     (if (<= y -1.15e-138)
       t_1
       (if (<= y 1.25e-204)
         (/ x (* y (exp b)))
         (if (<= y 3.1e+44) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(a, (t + -1.0)) * (x / y);
	double t_2 = (x * pow(z, y)) / y;
	double tmp;
	if (y <= -3.3e+40) {
		tmp = t_2;
	} else if (y <= -1.15e-138) {
		tmp = t_1;
	} else if (y <= 1.25e-204) {
		tmp = x / (y * exp(b));
	} else if (y <= 3.1e+44) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a ** (t + (-1.0d0))) * (x / y)
    t_2 = (x * (z ** y)) / y
    if (y <= (-3.3d+40)) then
        tmp = t_2
    else if (y <= (-1.15d-138)) then
        tmp = t_1
    else if (y <= 1.25d-204) then
        tmp = x / (y * exp(b))
    else if (y <= 3.1d+44) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.pow(a, (t + -1.0)) * (x / y);
	double t_2 = (x * Math.pow(z, y)) / y;
	double tmp;
	if (y <= -3.3e+40) {
		tmp = t_2;
	} else if (y <= -1.15e-138) {
		tmp = t_1;
	} else if (y <= 1.25e-204) {
		tmp = x / (y * Math.exp(b));
	} else if (y <= 3.1e+44) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(a, (t + -1.0)) * (x / y)
	t_2 = (x * math.pow(z, y)) / y
	tmp = 0
	if y <= -3.3e+40:
		tmp = t_2
	elif y <= -1.15e-138:
		tmp = t_1
	elif y <= 1.25e-204:
		tmp = x / (y * math.exp(b))
	elif y <= 3.1e+44:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64((a ^ Float64(t + -1.0)) * Float64(x / y))
	t_2 = Float64(Float64(x * (z ^ y)) / y)
	tmp = 0.0
	if (y <= -3.3e+40)
		tmp = t_2;
	elseif (y <= -1.15e-138)
		tmp = t_1;
	elseif (y <= 1.25e-204)
		tmp = Float64(x / Float64(y * exp(b)));
	elseif (y <= 3.1e+44)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a ^ (t + -1.0)) * (x / y);
	t_2 = (x * (z ^ y)) / y;
	tmp = 0.0;
	if (y <= -3.3e+40)
		tmp = t_2;
	elseif (y <= -1.15e-138)
		tmp = t_1;
	elseif (y <= 1.25e-204)
		tmp = x / (y * exp(b));
	elseif (y <= 3.1e+44)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -3.3e+40], t$95$2, If[LessEqual[y, -1.15e-138], t$95$1, If[LessEqual[y, 1.25e-204], N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+44], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2999999999999998e40 or 3.09999999999999996e44 < y

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 96.2

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 96.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot {z}^{y}}}{y} \]

      3. pow-lowering-pow.f64 88.6

         \[\leadsto \frac{x \cdot \color{blue}{{z}^{y}}}{y} \]

   8. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

   if -3.2999999999999998e40 < y < -1.14999999999999995e-138 or 1.25e-204 < y < 3.09999999999999996e44

   1. Initial program 98.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 81.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 81.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 67.4

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 67.4%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]

   if -1.14999999999999995e-138 < y < 1.25e-204

   1. Initial program 99.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 87.6

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 87.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 67.6

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 67.6%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]
   9. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{e^{b}}}}{y} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{e^{b}}}}{y} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      6. exp-lowering-exp.f64 67.6

         \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 67.6%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 64.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{{a}^{t}}{y}\\ \mathbf{if}\;t + -1 \leq -2 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t + -1 \leq 10000000000000:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (/ (pow a t) y))))
   (if (<= (+ t -1.0) -2e+104)
     t_1
     (if (<= (+ t -1.0) 10000000000000.0) (/ x (* y (exp b))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (pow(a, t) / y);
	double tmp;
	if ((t + -1.0) <= -2e+104) {
		tmp = t_1;
	} else if ((t + -1.0) <= 10000000000000.0) {
		tmp = x / (y * exp(b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((a ** t) / y)
    if ((t + (-1.0d0)) <= (-2d+104)) then
        tmp = t_1
    else if ((t + (-1.0d0)) <= 10000000000000.0d0) then
        tmp = x / (y * exp(b))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (Math.pow(a, t) / y);
	double tmp;
	if ((t + -1.0) <= -2e+104) {
		tmp = t_1;
	} else if ((t + -1.0) <= 10000000000000.0) {
		tmp = x / (y * Math.exp(b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * (math.pow(a, t) / y)
	tmp = 0
	if (t + -1.0) <= -2e+104:
		tmp = t_1
	elif (t + -1.0) <= 10000000000000.0:
		tmp = x / (y * math.exp(b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64((a ^ t) / y))
	tmp = 0.0
	if (Float64(t + -1.0) <= -2e+104)
		tmp = t_1;
	elseif (Float64(t + -1.0) <= 10000000000000.0)
		tmp = Float64(x / Float64(y * exp(b)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * ((a ^ t) / y);
	tmp = 0.0;
	if ((t + -1.0) <= -2e+104)
		tmp = t_1;
	elseif ((t + -1.0) <= 10000000000000.0)
		tmp = x / (y * exp(b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(N[Power[a, t], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t + -1.0), $MachinePrecision], -2e+104], t$95$1, If[LessEqual[N[(t + -1.0), $MachinePrecision], 10000000000000.0], N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 t #s(literal 1 binary64)) < -2e104 or 1e13 < (-.f64 t #s(literal 1 binary64))

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 92.1

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 92.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {a}^{t}}{y}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{{a}^{t}}{y}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{{a}^{t}}{y}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{{a}^{t}}{y}} \]

      4. pow-lowering-pow.f64 88.2

         \[\leadsto x \cdot \frac{\color{blue}{{a}^{t}}}{y} \]

   8. Simplified 88.2%

      \[\leadsto \color{blue}{x \cdot \frac{{a}^{t}}{y}} \]

   if -2e104 < (-.f64 t #s(literal 1 binary64)) < 1e13

   1. Initial program 98.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 62.4

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 62.4%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 61.8

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 61.8%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]
   9. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{e^{b}}}}{y} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{e^{b}}}}{y} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      6. exp-lowering-exp.f64 61.8

         \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 61.8%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t + -1 \leq -2 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \frac{{a}^{t}}{y}\\ \mathbf{elif}\;t + -1 \leq 10000000000000:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{a}^{t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y \cdot a}\\ t_2 := \frac{x}{y \cdot e^{b}}\\ \mathbf{if}\;b \leq -520000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq -3.4 \cdot 10^{-292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, 2, \frac{4 \cdot \left(\frac{x}{y \cdot \left(a \cdot b\right)} - t\_1\right)}{b}\right)}{b \cdot b}\\ \mathbf{elif}\;b \leq 0.062:\\ \;\;\;\;\frac{x}{b \cdot \left(a \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* y a))) (t_2 (/ x (* y (exp b)))))
   (if (<= b -520000000000.0)
     t_2
     (if (<= b -3.4e-292)
       (/ (fma t_1 2.0 (/ (* 4.0 (- (/ x (* y (* a b))) t_1)) b)) (* b b))
       (if (<= b 0.062) (/ x (* b (* a (+ y (/ y b))))) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * a);
	double t_2 = x / (y * exp(b));
	double tmp;
	if (b <= -520000000000.0) {
		tmp = t_2;
	} else if (b <= -3.4e-292) {
		tmp = fma(t_1, 2.0, ((4.0 * ((x / (y * (a * b))) - t_1)) / b)) / (b * b);
	} else if (b <= 0.062) {
		tmp = x / (b * (a * (y + (y / b))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(y * a))
	t_2 = Float64(x / Float64(y * exp(b)))
	tmp = 0.0
	if (b <= -520000000000.0)
		tmp = t_2;
	elseif (b <= -3.4e-292)
		tmp = Float64(fma(t_1, 2.0, Float64(Float64(4.0 * Float64(Float64(x / Float64(y * Float64(a * b))) - t_1)) / b)) / Float64(b * b));
	elseif (b <= 0.062)
		tmp = Float64(x / Float64(b * Float64(a * Float64(y + Float64(y / b)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -520000000000.0], t$95$2, If[LessEqual[b, -3.4e-292], N[(N[(t$95$1 * 2.0 + N[(N[(4.0 * N[(N[(x / N[(y * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.062], N[(x / N[(b * N[(a * N[(y + N[(y / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.2e11 or 0.062 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 90.3

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 90.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 79.3

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 79.3%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]
   9. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{e^{b}}}}{y} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{e^{b}}}}{y} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      6. exp-lowering-exp.f64 79.3

         \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 79.3%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   if -5.2e11 < b < -3.40000000000000017e-292

   1. Initial program 99.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 77.0

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 29.5

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 29.5%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)} + y\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right)\right) + y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b\right)}\right) + y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right)}\right) + y\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)}\right)\right) + y\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot y\right)}\right) + y\right)} \]

       8. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot y}\right) + y\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot \left(b + b \cdot \left(\frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       10. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{b \cdot 1} + b \cdot \left(\frac{1}{2} \cdot b\right)\right) + y\right)} \]

       11. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b \cdot \left(1 + \frac{1}{2} \cdot b\right), y\right)}} \]

       13. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + 1\right)}, y\right)} \]

       14. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b\right) + b \cdot 1}, y\right)} \]

       15. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{b}, y\right)} \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot b, b\right)}, y\right)} \]

       17. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{2}}, b\right), y\right)} \]

       18. *-lowering-*.f64 29.5

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot 0.5}, b\right), y\right)} \]

   11. Simplified 29.5%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{-8 \cdot \frac{x}{a \cdot y} + 4 \cdot \frac{x}{a \cdot y}}{{b}^{2}} + 2 \cdot \frac{x}{a \cdot y}\right) - 4 \cdot \frac{x}{a \cdot \left(b \cdot y\right)}}{{b}^{2}}} \]

   14. Simplified 52.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x}{y \cdot a}, 2, \frac{4 \cdot \left(\frac{x}{y \cdot \left(b \cdot a\right)} - \frac{x}{y \cdot a}\right)}{b}\right)}{b \cdot b}} \]

   if -3.40000000000000017e-292 < b < 0.062

   1. Initial program 97.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 71.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 43.0

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot b} + y\right)} \]

       5. accelerator-lowering-fma.f64 43.1

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   11. Simplified 43.1%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(y, b, y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot y + \frac{a \cdot y}{b}\right)}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot y + \frac{a \cdot y}{b}\right)}} \]

       2. associate-/l* N/A

          \[\leadsto \frac{x}{b \cdot \left(a \cdot y + \color{blue}{a \cdot \frac{y}{b}}\right)} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \frac{x}{b \cdot \left(a \cdot \color{blue}{\left(y + \frac{y}{b}\right)}\right)} \]

       6. /-lowering-/.f64 55.2

          \[\leadsto \frac{x}{b \cdot \left(a \cdot \left(y + \color{blue}{\frac{y}{b}}\right)\right)} \]

   14. Simplified 55.2%

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -520000000000:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;b \leq -3.4 \cdot 10^{-292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y \cdot a}, 2, \frac{4 \cdot \left(\frac{x}{y \cdot \left(a \cdot b\right)} - \frac{x}{y \cdot a}\right)}{b}\right)}{b \cdot b}\\ \mathbf{elif}\;b \leq 0.062:\\ \;\;\;\;\frac{x}{b \cdot \left(a \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{{a}^{t}}{y}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 58000000000000:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (/ (pow a t) y))))
   (if (<= t -1e+99)
     t_1
     (if (<= t 58000000000000.0) (/ x (* a (* y (exp b)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (pow(a, t) / y);
	double tmp;
	if (t <= -1e+99) {
		tmp = t_1;
	} else if (t <= 58000000000000.0) {
		tmp = x / (a * (y * exp(b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((a ** t) / y)
    if (t <= (-1d+99)) then
        tmp = t_1
    else if (t <= 58000000000000.0d0) then
        tmp = x / (a * (y * exp(b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (Math.pow(a, t) / y);
	double tmp;
	if (t <= -1e+99) {
		tmp = t_1;
	} else if (t <= 58000000000000.0) {
		tmp = x / (a * (y * Math.exp(b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * (math.pow(a, t) / y)
	tmp = 0
	if t <= -1e+99:
		tmp = t_1
	elif t <= 58000000000000.0:
		tmp = x / (a * (y * math.exp(b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64((a ^ t) / y))
	tmp = 0.0
	if (t <= -1e+99)
		tmp = t_1;
	elseif (t <= 58000000000000.0)
		tmp = Float64(x / Float64(a * Float64(y * exp(b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * ((a ^ t) / y);
	tmp = 0.0;
	if (t <= -1e+99)
		tmp = t_1;
	elseif (t <= 58000000000000.0)
		tmp = x / (a * (y * exp(b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(N[Power[a, t], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e+99], t$95$1, If[LessEqual[t, 58000000000000.0], N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.9999999999999997e98 or 5.8e13 < t

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 92.1

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 92.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {a}^{t}}{y}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{{a}^{t}}{y}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{{a}^{t}}{y}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{{a}^{t}}{y}} \]

      4. pow-lowering-pow.f64 88.2

         \[\leadsto x \cdot \frac{\color{blue}{{a}^{t}}}{y} \]

   8. Simplified 88.2%

      \[\leadsto \color{blue}{x \cdot \frac{{a}^{t}}{y}} \]

   if -9.9999999999999997e98 < t < 5.8e13

   1. Initial program 98.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 66.3

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 66.3%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 75.6

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 54.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y \cdot a}\\ t_2 := y + \frac{y}{b}\\ \mathbf{if}\;b \leq -8.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-294}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, 2, \frac{4 \cdot \left(\frac{x}{y \cdot \left(a \cdot b\right)} - t\_1\right)}{b}\right)}{b \cdot b}\\ \mathbf{elif}\;b \leq 3100:\\ \;\;\;\;\frac{x}{b \cdot \left(a \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\frac{\mathsf{fma}\left(y, 0.5, \frac{t\_2}{b}\right)}{b} - y \cdot -0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* y a))) (t_2 (+ y (/ y b))))
   (if (<= b -8.5e+80)
     (/ (* x (fma b (fma b (fma b -0.16666666666666666 0.5) -1.0) 1.0)) y)
     (if (<= b -1.4e-294)
       (/ (fma t_1 2.0 (/ (* 4.0 (- (/ x (* y (* a b))) t_1)) b)) (* b b))
       (if (<= b 3100.0)
         (/ x (* b (* a t_2)))
         (/
          x
          (*
           a
           (*
            (* b (* b b))
            (- (/ (fma y 0.5 (/ t_2 b)) b) (* y -0.16666666666666666))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * a);
	double t_2 = y + (y / b);
	double tmp;
	if (b <= -8.5e+80) {
		tmp = (x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y;
	} else if (b <= -1.4e-294) {
		tmp = fma(t_1, 2.0, ((4.0 * ((x / (y * (a * b))) - t_1)) / b)) / (b * b);
	} else if (b <= 3100.0) {
		tmp = x / (b * (a * t_2));
	} else {
		tmp = x / (a * ((b * (b * b)) * ((fma(y, 0.5, (t_2 / b)) / b) - (y * -0.16666666666666666))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(y * a))
	t_2 = Float64(y + Float64(y / b))
	tmp = 0.0
	if (b <= -8.5e+80)
		tmp = Float64(Float64(x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y);
	elseif (b <= -1.4e-294)
		tmp = Float64(fma(t_1, 2.0, Float64(Float64(4.0 * Float64(Float64(x / Float64(y * Float64(a * b))) - t_1)) / b)) / Float64(b * b));
	elseif (b <= 3100.0)
		tmp = Float64(x / Float64(b * Float64(a * t_2)));
	else
		tmp = Float64(x / Float64(a * Float64(Float64(b * Float64(b * b)) * Float64(Float64(fma(y, 0.5, Float64(t_2 / b)) / b) - Float64(y * -0.16666666666666666)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y + N[(y / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.5e+80], N[(N[(x * N[(b * N[(b * N[(b * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, -1.4e-294], N[(N[(t$95$1 * 2.0 + N[(N[(4.0 * N[(N[(x / N[(y * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3100.0], N[(x / N[(b * N[(a * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * 0.5 + N[(t$95$2 / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] - N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.50000000000000007e80

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 98.2

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 98.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 92.7

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 92.7%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right) + 1\right)}}{y} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1, 1\right)}}{y} \]

       3. sub-neg N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{y} \]

       4. metadata-eval N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \color{blue}{-1}, 1\right)}{y} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{-1}{6} \cdot b, -1\right)}, 1\right)}{y} \]

       6. +-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot b + \frac{1}{2}}, -1\right), 1\right)}{y} \]

       7. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{y} \]

       8. accelerator-lowering-fma.f64 89.1

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{y} \]

   11. Simplified 89.1%

       \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{y} \]

   if -8.50000000000000007e80 < b < -1.39999999999999995e-294

   1. Initial program 99.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 67.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 33.1

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 33.1%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)} + y\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right)\right) + y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b\right)}\right) + y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right)}\right) + y\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)}\right)\right) + y\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot y\right)}\right) + y\right)} \]

       8. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot y}\right) + y\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot \left(b + b \cdot \left(\frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       10. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{b \cdot 1} + b \cdot \left(\frac{1}{2} \cdot b\right)\right) + y\right)} \]

       11. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b \cdot \left(1 + \frac{1}{2} \cdot b\right), y\right)}} \]

       13. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + 1\right)}, y\right)} \]

       14. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b\right) + b \cdot 1}, y\right)} \]

       15. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{b}, y\right)} \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot b, b\right)}, y\right)} \]

       17. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{2}}, b\right), y\right)} \]

       18. *-lowering-*.f64 31.8

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot 0.5}, b\right), y\right)} \]

   11. Simplified 31.8%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{-8 \cdot \frac{x}{a \cdot y} + 4 \cdot \frac{x}{a \cdot y}}{{b}^{2}} + 2 \cdot \frac{x}{a \cdot y}\right) - 4 \cdot \frac{x}{a \cdot \left(b \cdot y\right)}}{{b}^{2}}} \]

   14. Simplified 47.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x}{y \cdot a}, 2, \frac{4 \cdot \left(\frac{x}{y \cdot \left(b \cdot a\right)} - \frac{x}{y \cdot a}\right)}{b}\right)}{b \cdot b}} \]

   if -1.39999999999999995e-294 < b < 3100

   1. Initial program 97.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 69.9

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 41.2

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 41.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot b} + y\right)} \]

       5. accelerator-lowering-fma.f64 41.3

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   11. Simplified 41.3%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(y, b, y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot y + \frac{a \cdot y}{b}\right)}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot y + \frac{a \cdot y}{b}\right)}} \]

       2. associate-/l* N/A

          \[\leadsto \frac{x}{b \cdot \left(a \cdot y + \color{blue}{a \cdot \frac{y}{b}}\right)} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \frac{x}{b \cdot \left(a \cdot \color{blue}{\left(y + \frac{y}{b}\right)}\right)} \]

       6. /-lowering-/.f64 52.8

          \[\leadsto \frac{x}{b \cdot \left(a \cdot \left(y + \color{blue}{\frac{y}{b}}\right)\right)} \]

   14. Simplified 52.8%

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

   if 3100 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 63.9

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 80.0

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 80.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right) + \frac{1}{2} \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right) + \frac{1}{2} \cdot y\right)\right) + y\right)}} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, y + b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right) + \frac{1}{2} \cdot y\right), y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right) + \frac{1}{2} \cdot y\right) + y}, y\right)} \]

       4. distribute-lft-in N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right)\right) + b \cdot \left(\frac{1}{2} \cdot y\right)\right)} + y, y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \left(b \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot b\right) \cdot y\right)} + b \cdot \left(\frac{1}{2} \cdot y\right)\right) + y, y\right)} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \left(\color{blue}{\left(b \cdot \left(\frac{1}{6} \cdot b\right)\right) \cdot y} + b \cdot \left(\frac{1}{2} \cdot y\right)\right) + y, y\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \left(\left(b \cdot \left(\frac{1}{6} \cdot b\right)\right) \cdot y + \color{blue}{\left(b \cdot \frac{1}{2}\right) \cdot y}\right) + y, y\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \left(\left(b \cdot \left(\frac{1}{6} \cdot b\right)\right) \cdot y + \color{blue}{\left(\frac{1}{2} \cdot b\right)} \cdot y\right) + y, y\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{y \cdot \left(b \cdot \left(\frac{1}{6} \cdot b\right) + \frac{1}{2} \cdot b\right)} + y, y\right)} \]

       10. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, y \cdot \left(b \cdot \left(\frac{1}{6} \cdot b\right) + \color{blue}{b \cdot \frac{1}{2}}\right) + y, y\right)} \]

       11. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, y \cdot \color{blue}{\left(b \cdot \left(\frac{1}{6} \cdot b + \frac{1}{2}\right)\right)} + y, y\right)} \]

       12. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, y \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right) + y, y\right)} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(y, b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), y\right)}, y\right)} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}, y\right), y\right)} \]

       15. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot \color{blue}{\left(\frac{1}{6} \cdot b + \frac{1}{2}\right)}, y\right), y\right)} \]

       16. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot \left(\color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}\right), y\right), y\right)} \]

       17. accelerator-lowering-fma.f64 66.2

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, y\right), y\right)} \]

   11. Simplified 66.2%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), y\right), y\right)}} \]

   12. Taylor expanded in b around -inf

       \[\leadsto \frac{x}{a \cdot \color{blue}{\left(-1 \cdot \left({b}^{3} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot y + -1 \cdot \frac{y}{b}}{b} + \frac{1}{2} \cdot y}{b} + \frac{-1}{6} \cdot y\right)\right)\right)}} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(-1 \cdot {b}^{3}\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot y + -1 \cdot \frac{y}{b}}{b} + \frac{1}{2} \cdot y}{b} + \frac{-1}{6} \cdot y\right)\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot y + -1 \cdot \frac{y}{b}}{b} + \frac{1}{2} \cdot y}{b} + \frac{-1}{6} \cdot y\right) \cdot \left(-1 \cdot {b}^{3}\right)\right)}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot y + -1 \cdot \frac{y}{b}}{b} + \frac{1}{2} \cdot y}{b} + \frac{-1}{6} \cdot y\right) \cdot \left(-1 \cdot {b}^{3}\right)\right)}} \]

   14. Simplified 67.5%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(y \cdot -0.16666666666666666 - \frac{\mathsf{fma}\left(y, 0.5, \frac{y + \frac{y}{b}}{b}\right)}{b}\right) \cdot \left(-b \cdot \left(b \cdot b\right)\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-294}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y \cdot a}, 2, \frac{4 \cdot \left(\frac{x}{y \cdot \left(a \cdot b\right)} - \frac{x}{y \cdot a}\right)}{b}\right)}{b \cdot b}\\ \mathbf{elif}\;b \leq 3100:\\ \;\;\;\;\frac{x}{b \cdot \left(a \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\frac{\mathsf{fma}\left(y, 0.5, \frac{y + \frac{y}{b}}{b}\right)}{b} - y \cdot -0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \frac{y}{b}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-297}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;b \leq 3600:\\ \;\;\;\;\frac{x}{b \cdot \left(a \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\frac{\mathsf{fma}\left(y, 0.5, \frac{t\_1}{b}\right)}{b} - y \cdot -0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (/ y b))))
   (if (<= b -2.8e+75)
     (/ (* x (fma b (fma b (fma b -0.16666666666666666 0.5) -1.0) 1.0)) y)
     (if (<= b -1.35e-297)
       (/ (* x 2.0) (* y (* a (* b b))))
       (if (<= b 3600.0)
         (/ x (* b (* a t_1)))
         (/
          x
          (*
           a
           (*
            (* b (* b b))
            (- (/ (fma y 0.5 (/ t_1 b)) b) (* y -0.16666666666666666))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (y / b);
	double tmp;
	if (b <= -2.8e+75) {
		tmp = (x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y;
	} else if (b <= -1.35e-297) {
		tmp = (x * 2.0) / (y * (a * (b * b)));
	} else if (b <= 3600.0) {
		tmp = x / (b * (a * t_1));
	} else {
		tmp = x / (a * ((b * (b * b)) * ((fma(y, 0.5, (t_1 / b)) / b) - (y * -0.16666666666666666))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(y / b))
	tmp = 0.0
	if (b <= -2.8e+75)
		tmp = Float64(Float64(x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y);
	elseif (b <= -1.35e-297)
		tmp = Float64(Float64(x * 2.0) / Float64(y * Float64(a * Float64(b * b))));
	elseif (b <= 3600.0)
		tmp = Float64(x / Float64(b * Float64(a * t_1)));
	else
		tmp = Float64(x / Float64(a * Float64(Float64(b * Float64(b * b)) * Float64(Float64(fma(y, 0.5, Float64(t_1 / b)) / b) - Float64(y * -0.16666666666666666)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(y / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.8e+75], N[(N[(x * N[(b * N[(b * N[(b * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, -1.35e-297], N[(N[(x * 2.0), $MachinePrecision] / N[(y * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3600.0], N[(x / N[(b * N[(a * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * 0.5 + N[(t$95$1 / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] - N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.80000000000000012e75

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 98.2

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 98.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 92.7

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 92.7%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right) + 1\right)}}{y} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1, 1\right)}}{y} \]

       3. sub-neg N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{y} \]

       4. metadata-eval N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \color{blue}{-1}, 1\right)}{y} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{-1}{6} \cdot b, -1\right)}, 1\right)}{y} \]

       6. +-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot b + \frac{1}{2}}, -1\right), 1\right)}{y} \]

       7. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{y} \]

       8. accelerator-lowering-fma.f64 89.1

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{y} \]

   11. Simplified 89.1%

       \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{y} \]

   if -2.80000000000000012e75 < b < -1.3500000000000001e-297

   1. Initial program 99.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 67.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 33.1

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 33.1%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)} + y\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right)\right) + y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b\right)}\right) + y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right)}\right) + y\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)}\right)\right) + y\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot y\right)}\right) + y\right)} \]

       8. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot y}\right) + y\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot \left(b + b \cdot \left(\frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       10. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{b \cdot 1} + b \cdot \left(\frac{1}{2} \cdot b\right)\right) + y\right)} \]

       11. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b \cdot \left(1 + \frac{1}{2} \cdot b\right), y\right)}} \]

       13. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + 1\right)}, y\right)} \]

       14. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b\right) + b \cdot 1}, y\right)} \]

       15. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{b}, y\right)} \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot b, b\right)}, y\right)} \]

       17. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{2}}, b\right), y\right)} \]

       18. *-lowering-*.f64 31.8

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot 0.5}, b\right), y\right)} \]

   11. Simplified 31.8%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{a \cdot \left({b}^{2} \cdot y\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(a \cdot {b}^{2}\right) \cdot y}} \]

       6. *-commutative N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot \left(a \cdot {b}^{2}\right)}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot \left(a \cdot {b}^{2}\right)}} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{y \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{x \cdot 2}{y \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

       10. *-lowering-*.f64 46.1

           \[\leadsto \frac{x \cdot 2}{y \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

   14. Simplified 46.1%

       \[\leadsto \color{blue}{\frac{x \cdot 2}{y \cdot \left(a \cdot \left(b \cdot b\right)\right)}} \]

   if -1.3500000000000001e-297 < b < 3600

   1. Initial program 97.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 69.9

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 41.2

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 41.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot b} + y\right)} \]

       5. accelerator-lowering-fma.f64 41.3

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   11. Simplified 41.3%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(y, b, y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot y + \frac{a \cdot y}{b}\right)}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot y + \frac{a \cdot y}{b}\right)}} \]

       2. associate-/l* N/A

          \[\leadsto \frac{x}{b \cdot \left(a \cdot y + \color{blue}{a \cdot \frac{y}{b}}\right)} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \frac{x}{b \cdot \left(a \cdot \color{blue}{\left(y + \frac{y}{b}\right)}\right)} \]

       6. /-lowering-/.f64 52.8

          \[\leadsto \frac{x}{b \cdot \left(a \cdot \left(y + \color{blue}{\frac{y}{b}}\right)\right)} \]

   14. Simplified 52.8%

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

   if 3600 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 63.9

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 80.0

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 80.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right) + \frac{1}{2} \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right) + \frac{1}{2} \cdot y\right)\right) + y\right)}} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, y + b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right) + \frac{1}{2} \cdot y\right), y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right) + \frac{1}{2} \cdot y\right) + y}, y\right)} \]

       4. distribute-lft-in N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right)\right) + b \cdot \left(\frac{1}{2} \cdot y\right)\right)} + y, y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \left(b \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot b\right) \cdot y\right)} + b \cdot \left(\frac{1}{2} \cdot y\right)\right) + y, y\right)} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \left(\color{blue}{\left(b \cdot \left(\frac{1}{6} \cdot b\right)\right) \cdot y} + b \cdot \left(\frac{1}{2} \cdot y\right)\right) + y, y\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \left(\left(b \cdot \left(\frac{1}{6} \cdot b\right)\right) \cdot y + \color{blue}{\left(b \cdot \frac{1}{2}\right) \cdot y}\right) + y, y\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \left(\left(b \cdot \left(\frac{1}{6} \cdot b\right)\right) \cdot y + \color{blue}{\left(\frac{1}{2} \cdot b\right)} \cdot y\right) + y, y\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{y \cdot \left(b \cdot \left(\frac{1}{6} \cdot b\right) + \frac{1}{2} \cdot b\right)} + y, y\right)} \]

       10. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, y \cdot \left(b \cdot \left(\frac{1}{6} \cdot b\right) + \color{blue}{b \cdot \frac{1}{2}}\right) + y, y\right)} \]

       11. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, y \cdot \color{blue}{\left(b \cdot \left(\frac{1}{6} \cdot b + \frac{1}{2}\right)\right)} + y, y\right)} \]

       12. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, y \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right) + y, y\right)} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(y, b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), y\right)}, y\right)} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}, y\right), y\right)} \]

       15. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot \color{blue}{\left(\frac{1}{6} \cdot b + \frac{1}{2}\right)}, y\right), y\right)} \]

       16. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot \left(\color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}\right), y\right), y\right)} \]

       17. accelerator-lowering-fma.f64 66.2

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, y\right), y\right)} \]

   11. Simplified 66.2%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), y\right), y\right)}} \]

   12. Taylor expanded in b around -inf

       \[\leadsto \frac{x}{a \cdot \color{blue}{\left(-1 \cdot \left({b}^{3} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot y + -1 \cdot \frac{y}{b}}{b} + \frac{1}{2} \cdot y}{b} + \frac{-1}{6} \cdot y\right)\right)\right)}} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(-1 \cdot {b}^{3}\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot y + -1 \cdot \frac{y}{b}}{b} + \frac{1}{2} \cdot y}{b} + \frac{-1}{6} \cdot y\right)\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot y + -1 \cdot \frac{y}{b}}{b} + \frac{1}{2} \cdot y}{b} + \frac{-1}{6} \cdot y\right) \cdot \left(-1 \cdot {b}^{3}\right)\right)}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot y + -1 \cdot \frac{y}{b}}{b} + \frac{1}{2} \cdot y}{b} + \frac{-1}{6} \cdot y\right) \cdot \left(-1 \cdot {b}^{3}\right)\right)}} \]

   14. Simplified 67.5%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(y \cdot -0.16666666666666666 - \frac{\mathsf{fma}\left(y, 0.5, \frac{y + \frac{y}{b}}{b}\right)}{b}\right) \cdot \left(-b \cdot \left(b \cdot b\right)\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-297}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;b \leq 3600:\\ \;\;\;\;\frac{x}{b \cdot \left(a \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\frac{\mathsf{fma}\left(y, 0.5, \frac{y + \frac{y}{b}}{b}\right)}{b} - y \cdot -0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq -7.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;b \leq 3300:\\ \;\;\;\;\frac{x}{b \cdot \left(a \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -3.5e+75)
   (/ (* x (fma b (fma b (fma b -0.16666666666666666 0.5) -1.0) 1.0)) y)
   (if (<= b -7.5e-297)
     (/ (* x 2.0) (* y (* a (* b b))))
     (if (<= b 3300.0)
       (/ x (* b (* a (+ y (/ y b)))))
       (/
        x
        (*
         a
         (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.5e+75) {
		tmp = (x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y;
	} else if (b <= -7.5e-297) {
		tmp = (x * 2.0) / (y * (a * (b * b)));
	} else if (b <= 3300.0) {
		tmp = x / (b * (a * (y + (y / b))));
	} else {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -3.5e+75)
		tmp = Float64(Float64(x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y);
	elseif (b <= -7.5e-297)
		tmp = Float64(Float64(x * 2.0) / Float64(y * Float64(a * Float64(b * b))));
	elseif (b <= 3300.0)
		tmp = Float64(x / Float64(b * Float64(a * Float64(y + Float64(y / b)))));
	else
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -3.5e+75], N[(N[(x * N[(b * N[(b * N[(b * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, -7.5e-297], N[(N[(x * 2.0), $MachinePrecision] / N[(y * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3300.0], N[(x / N[(b * N[(a * N[(y + N[(y / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.4999999999999998e75

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 98.2

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 98.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 92.7

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 92.7%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right) + 1\right)}}{y} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1, 1\right)}}{y} \]

       3. sub-neg N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{y} \]

       4. metadata-eval N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \color{blue}{-1}, 1\right)}{y} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{-1}{6} \cdot b, -1\right)}, 1\right)}{y} \]

       6. +-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot b + \frac{1}{2}}, -1\right), 1\right)}{y} \]

       7. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{y} \]

       8. accelerator-lowering-fma.f64 89.1

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{y} \]

   11. Simplified 89.1%

       \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{y} \]

   if -3.4999999999999998e75 < b < -7.4999999999999994e-297

   1. Initial program 99.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 67.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 33.1

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 33.1%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)} + y\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right)\right) + y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b\right)}\right) + y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right)}\right) + y\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)}\right)\right) + y\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot y\right)}\right) + y\right)} \]

       8. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot y}\right) + y\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot \left(b + b \cdot \left(\frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       10. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{b \cdot 1} + b \cdot \left(\frac{1}{2} \cdot b\right)\right) + y\right)} \]

       11. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b \cdot \left(1 + \frac{1}{2} \cdot b\right), y\right)}} \]

       13. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + 1\right)}, y\right)} \]

       14. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b\right) + b \cdot 1}, y\right)} \]

       15. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{b}, y\right)} \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot b, b\right)}, y\right)} \]

       17. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{2}}, b\right), y\right)} \]

       18. *-lowering-*.f64 31.8

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot 0.5}, b\right), y\right)} \]

   11. Simplified 31.8%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{a \cdot \left({b}^{2} \cdot y\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(a \cdot {b}^{2}\right) \cdot y}} \]

       6. *-commutative N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot \left(a \cdot {b}^{2}\right)}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot \left(a \cdot {b}^{2}\right)}} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{y \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{x \cdot 2}{y \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

       10. *-lowering-*.f64 46.1

           \[\leadsto \frac{x \cdot 2}{y \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

   14. Simplified 46.1%

       \[\leadsto \color{blue}{\frac{x \cdot 2}{y \cdot \left(a \cdot \left(b \cdot b\right)\right)}} \]

   if -7.4999999999999994e-297 < b < 3300

   1. Initial program 97.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 69.9

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 41.2

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 41.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot b} + y\right)} \]

       5. accelerator-lowering-fma.f64 41.3

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   11. Simplified 41.3%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(y, b, y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot y + \frac{a \cdot y}{b}\right)}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot y + \frac{a \cdot y}{b}\right)}} \]

       2. associate-/l* N/A

          \[\leadsto \frac{x}{b \cdot \left(a \cdot y + \color{blue}{a \cdot \frac{y}{b}}\right)} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \frac{x}{b \cdot \left(a \cdot \color{blue}{\left(y + \frac{y}{b}\right)}\right)} \]

       6. /-lowering-/.f64 52.8

          \[\leadsto \frac{x}{b \cdot \left(a \cdot \left(y + \color{blue}{\frac{y}{b}}\right)\right)} \]

   14. Simplified 52.8%

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

   if 3300 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 63.9

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 80.0

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 80.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. accelerator-lowering-fma.f64 67.5

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 67.5%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 53.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-292}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;b \leq 0.032:\\ \;\;\;\;\frac{x}{b \cdot \left(a \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(0.16666666666666666 \cdot \left(b \cdot \left(y \cdot \left(b \cdot b\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -2.9e+75)
   (/ (* x (fma b (fma b (fma b -0.16666666666666666 0.5) -1.0) 1.0)) y)
   (if (<= b -2.6e-292)
     (/ (* x 2.0) (* y (* a (* b b))))
     (if (<= b 0.032)
       (/ x (* b (* a (+ y (/ y b)))))
       (/ x (* a (* 0.16666666666666666 (* b (* y (* b b))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.9e+75) {
		tmp = (x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y;
	} else if (b <= -2.6e-292) {
		tmp = (x * 2.0) / (y * (a * (b * b)));
	} else if (b <= 0.032) {
		tmp = x / (b * (a * (y + (y / b))));
	} else {
		tmp = x / (a * (0.16666666666666666 * (b * (y * (b * b)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -2.9e+75)
		tmp = Float64(Float64(x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y);
	elseif (b <= -2.6e-292)
		tmp = Float64(Float64(x * 2.0) / Float64(y * Float64(a * Float64(b * b))));
	elseif (b <= 0.032)
		tmp = Float64(x / Float64(b * Float64(a * Float64(y + Float64(y / b)))));
	else
		tmp = Float64(x / Float64(a * Float64(0.16666666666666666 * Float64(b * Float64(y * Float64(b * b))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -2.9e+75], N[(N[(x * N[(b * N[(b * N[(b * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, -2.6e-292], N[(N[(x * 2.0), $MachinePrecision] / N[(y * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.032], N[(x / N[(b * N[(a * N[(y + N[(y / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(0.16666666666666666 * N[(b * N[(y * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.8999999999999998e75

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 98.2

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 98.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 92.7

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 92.7%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right) + 1\right)}}{y} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1, 1\right)}}{y} \]

       3. sub-neg N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{y} \]

       4. metadata-eval N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \color{blue}{-1}, 1\right)}{y} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{-1}{6} \cdot b, -1\right)}, 1\right)}{y} \]

       6. +-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot b + \frac{1}{2}}, -1\right), 1\right)}{y} \]

       7. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{y} \]

       8. accelerator-lowering-fma.f64 89.1

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{y} \]

   11. Simplified 89.1%

       \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{y} \]

   if -2.8999999999999998e75 < b < -2.60000000000000013e-292

   1. Initial program 99.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 67.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 33.1

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 33.1%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)} + y\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right)\right) + y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b\right)}\right) + y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right)}\right) + y\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)}\right)\right) + y\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot y\right)}\right) + y\right)} \]

       8. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot y}\right) + y\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot \left(b + b \cdot \left(\frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       10. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{b \cdot 1} + b \cdot \left(\frac{1}{2} \cdot b\right)\right) + y\right)} \]

       11. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b \cdot \left(1 + \frac{1}{2} \cdot b\right), y\right)}} \]

       13. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + 1\right)}, y\right)} \]

       14. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b\right) + b \cdot 1}, y\right)} \]

       15. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{b}, y\right)} \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot b, b\right)}, y\right)} \]

       17. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{2}}, b\right), y\right)} \]

       18. *-lowering-*.f64 31.8

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot 0.5}, b\right), y\right)} \]

   11. Simplified 31.8%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{a \cdot \left({b}^{2} \cdot y\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(a \cdot {b}^{2}\right) \cdot y}} \]

       6. *-commutative N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot \left(a \cdot {b}^{2}\right)}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot \left(a \cdot {b}^{2}\right)}} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{y \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{x \cdot 2}{y \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

       10. *-lowering-*.f64 46.1

           \[\leadsto \frac{x \cdot 2}{y \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

   14. Simplified 46.1%

       \[\leadsto \color{blue}{\frac{x \cdot 2}{y \cdot \left(a \cdot \left(b \cdot b\right)\right)}} \]

   if -2.60000000000000013e-292 < b < 0.032000000000000001

   1. Initial program 97.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 71.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 43.0

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot b} + y\right)} \]

       5. accelerator-lowering-fma.f64 43.1

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   11. Simplified 43.1%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(y, b, y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot y + \frac{a \cdot y}{b}\right)}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot y + \frac{a \cdot y}{b}\right)}} \]

       2. associate-/l* N/A

          \[\leadsto \frac{x}{b \cdot \left(a \cdot y + \color{blue}{a \cdot \frac{y}{b}}\right)} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \frac{x}{b \cdot \left(a \cdot \color{blue}{\left(y + \frac{y}{b}\right)}\right)} \]

       6. /-lowering-/.f64 55.2

          \[\leadsto \frac{x}{b \cdot \left(a \cdot \left(y + \color{blue}{\frac{y}{b}}\right)\right)} \]

   14. Simplified 55.2%

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

   if 0.032000000000000001 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 62.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 62.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 76.8

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right) + \frac{1}{2} \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right) + \frac{1}{2} \cdot y\right)\right) + y\right)}} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, y + b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right) + \frac{1}{2} \cdot y\right), y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right) + \frac{1}{2} \cdot y\right) + y}, y\right)} \]

       4. distribute-lft-in N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right)\right) + b \cdot \left(\frac{1}{2} \cdot y\right)\right)} + y, y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \left(b \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot b\right) \cdot y\right)} + b \cdot \left(\frac{1}{2} \cdot y\right)\right) + y, y\right)} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \left(\color{blue}{\left(b \cdot \left(\frac{1}{6} \cdot b\right)\right) \cdot y} + b \cdot \left(\frac{1}{2} \cdot y\right)\right) + y, y\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \left(\left(b \cdot \left(\frac{1}{6} \cdot b\right)\right) \cdot y + \color{blue}{\left(b \cdot \frac{1}{2}\right) \cdot y}\right) + y, y\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \left(\left(b \cdot \left(\frac{1}{6} \cdot b\right)\right) \cdot y + \color{blue}{\left(\frac{1}{2} \cdot b\right)} \cdot y\right) + y, y\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{y \cdot \left(b \cdot \left(\frac{1}{6} \cdot b\right) + \frac{1}{2} \cdot b\right)} + y, y\right)} \]

       10. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, y \cdot \left(b \cdot \left(\frac{1}{6} \cdot b\right) + \color{blue}{b \cdot \frac{1}{2}}\right) + y, y\right)} \]

       11. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, y \cdot \color{blue}{\left(b \cdot \left(\frac{1}{6} \cdot b + \frac{1}{2}\right)\right)} + y, y\right)} \]

       12. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, y \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right) + y, y\right)} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(y, b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), y\right)}, y\right)} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}, y\right), y\right)} \]

       15. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot \color{blue}{\left(\frac{1}{6} \cdot b + \frac{1}{2}\right)}, y\right), y\right)} \]

       16. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot \left(\color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}\right), y\right), y\right)} \]

       17. accelerator-lowering-fma.f64 63.6

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, y\right), y\right)} \]

   11. Simplified 63.6%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(y, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), y\right), y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{\color{blue}{\frac{1}{6} \cdot \left(a \cdot \left({b}^{3} \cdot y\right)\right)}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{\left(a \cdot \left({b}^{3} \cdot y\right)\right) \cdot \frac{1}{6}}} \]

       2. associate-*l* N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(\left({b}^{3} \cdot y\right) \cdot \frac{1}{6}\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({b}^{3} \cdot y\right)\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot {b}^{3}\right)}\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot y\right) \cdot {b}^{3}\right)}} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(\left(\frac{1}{6} \cdot y\right) \cdot {b}^{3}\right)}} \]

       7. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(y \cdot {b}^{3}\right)\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({b}^{3} \cdot y\right)}\right)} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({b}^{3} \cdot y\right)\right)}} \]

       10. cube-mult N/A

           \[\leadsto \frac{x}{a \cdot \left(\frac{1}{6} \cdot \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot y\right)\right)} \]

       11. unpow2 N/A

           \[\leadsto \frac{x}{a \cdot \left(\frac{1}{6} \cdot \left(\left(b \cdot \color{blue}{{b}^{2}}\right) \cdot y\right)\right)} \]

       12. associate-*l* N/A

           \[\leadsto \frac{x}{a \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(b \cdot \left({b}^{2} \cdot y\right)\right)}\right)} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{a \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(b \cdot \left({b}^{2} \cdot y\right)\right)}\right)} \]

       14. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \left(\frac{1}{6} \cdot \left(b \cdot \color{blue}{\left(y \cdot {b}^{2}\right)}\right)\right)} \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{a \cdot \left(\frac{1}{6} \cdot \left(b \cdot \color{blue}{\left(y \cdot {b}^{2}\right)}\right)\right)} \]

       16. unpow2 N/A

           \[\leadsto \frac{x}{a \cdot \left(\frac{1}{6} \cdot \left(b \cdot \left(y \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)} \]

       17. *-lowering-*.f64 63.6

           \[\leadsto \frac{x}{a \cdot \left(0.16666666666666666 \cdot \left(b \cdot \left(y \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)} \]

   14. Simplified 63.6%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \left(0.16666666666666666 \cdot \left(b \cdot \left(y \cdot \left(b \cdot b\right)\right)\right)\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 53.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq -1.3 \cdot 10^{-291}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{b \cdot \left(a \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -3.5e+75)
   (/ (* x (fma b (fma b (fma b -0.16666666666666666 0.5) -1.0) 1.0)) y)
   (if (<= b -1.3e-291)
     (/ (* x 2.0) (* y (* a (* b b))))
     (if (<= b 7.2e+37)
       (/ x (* b (* a (+ y (/ y b)))))
       (/ x (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.5e+75) {
		tmp = (x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y;
	} else if (b <= -1.3e-291) {
		tmp = (x * 2.0) / (y * (a * (b * b)));
	} else if (b <= 7.2e+37) {
		tmp = x / (b * (a * (y + (y / b))));
	} else {
		tmp = x / (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -3.5e+75)
		tmp = Float64(Float64(x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y);
	elseif (b <= -1.3e-291)
		tmp = Float64(Float64(x * 2.0) / Float64(y * Float64(a * Float64(b * b))));
	elseif (b <= 7.2e+37)
		tmp = Float64(x / Float64(b * Float64(a * Float64(y + Float64(y / b)))));
	else
		tmp = Float64(x / Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -3.5e+75], N[(N[(x * N[(b * N[(b * N[(b * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, -1.3e-291], N[(N[(x * 2.0), $MachinePrecision] / N[(y * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e+37], N[(x / N[(b * N[(a * N[(y + N[(y / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.4999999999999998e75

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 98.2

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 98.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 92.7

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 92.7%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right) + 1\right)}}{y} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1, 1\right)}}{y} \]

       3. sub-neg N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{y} \]

       4. metadata-eval N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \color{blue}{-1}, 1\right)}{y} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{-1}{6} \cdot b, -1\right)}, 1\right)}{y} \]

       6. +-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot b + \frac{1}{2}}, -1\right), 1\right)}{y} \]

       7. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{y} \]

       8. accelerator-lowering-fma.f64 89.1

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{y} \]

   11. Simplified 89.1%

       \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{y} \]

   if -3.4999999999999998e75 < b < -1.2999999999999999e-291

   1. Initial program 99.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 67.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 33.1

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 33.1%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)} + y\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right)\right) + y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b\right)}\right) + y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right)}\right) + y\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)}\right)\right) + y\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot y\right)}\right) + y\right)} \]

       8. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot y}\right) + y\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot \left(b + b \cdot \left(\frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       10. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{b \cdot 1} + b \cdot \left(\frac{1}{2} \cdot b\right)\right) + y\right)} \]

       11. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b \cdot \left(1 + \frac{1}{2} \cdot b\right), y\right)}} \]

       13. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + 1\right)}, y\right)} \]

       14. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b\right) + b \cdot 1}, y\right)} \]

       15. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{b}, y\right)} \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot b, b\right)}, y\right)} \]

       17. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{2}}, b\right), y\right)} \]

       18. *-lowering-*.f64 31.8

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot 0.5}, b\right), y\right)} \]

   11. Simplified 31.8%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{a \cdot \left({b}^{2} \cdot y\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(a \cdot {b}^{2}\right) \cdot y}} \]

       6. *-commutative N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot \left(a \cdot {b}^{2}\right)}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot \left(a \cdot {b}^{2}\right)}} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{y \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{x \cdot 2}{y \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

       10. *-lowering-*.f64 46.1

           \[\leadsto \frac{x \cdot 2}{y \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

   14. Simplified 46.1%

       \[\leadsto \color{blue}{\frac{x \cdot 2}{y \cdot \left(a \cdot \left(b \cdot b\right)\right)}} \]

   if -1.2999999999999999e-291 < b < 7.19999999999999995e37

   1. Initial program 97.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 68.1

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 46.0

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 46.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot b} + y\right)} \]

       5. accelerator-lowering-fma.f64 37.3

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   11. Simplified 37.3%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(y, b, y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot y + \frac{a \cdot y}{b}\right)}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot y + \frac{a \cdot y}{b}\right)}} \]

       2. associate-/l* N/A

          \[\leadsto \frac{x}{b \cdot \left(a \cdot y + \color{blue}{a \cdot \frac{y}{b}}\right)} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \frac{x}{b \cdot \left(a \cdot \color{blue}{\left(y + \frac{y}{b}\right)}\right)} \]

       6. /-lowering-/.f64 46.7

          \[\leadsto \frac{x}{b \cdot \left(a \cdot \left(y + \color{blue}{\frac{y}{b}}\right)\right)} \]

   14. Simplified 46.7%

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(a \cdot \left(y + \frac{y}{b}\right)\right)}} \]

   if 7.19999999999999995e37 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 90.9

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 90.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 83.6

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 83.6%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]
   9. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{e^{b}}}}{y} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{e^{b}}}}{y} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      6. exp-lowering-exp.f64 83.6

         \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 83.6%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)} \]

       7. accelerator-lowering-fma.f64 74.7

          \[\leadsto \frac{x}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)} \]

   13. Simplified 74.7%

       \[\leadsto \frac{x}{y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 47.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, b \cdot \mathsf{fma}\left(b, 0.5, -1\right), x\right)}{y}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+113}:\\ \;\;\;\;\frac{x}{b \cdot \left(b \cdot \left(\left(y \cdot a\right) \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -2.4e+76)
   (/ (fma x (* b (fma b 0.5 -1.0)) x) y)
   (if (<= b 3.6e+113)
     (/ x (* b (* b (* (* y a) 0.5))))
     (/ x (fma y (fma b (* b 0.5) b) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.4e+76) {
		tmp = fma(x, (b * fma(b, 0.5, -1.0)), x) / y;
	} else if (b <= 3.6e+113) {
		tmp = x / (b * (b * ((y * a) * 0.5)));
	} else {
		tmp = x / fma(y, fma(b, (b * 0.5), b), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -2.4e+76)
		tmp = Float64(fma(x, Float64(b * fma(b, 0.5, -1.0)), x) / y);
	elseif (b <= 3.6e+113)
		tmp = Float64(x / Float64(b * Float64(b * Float64(Float64(y * a) * 0.5))));
	else
		tmp = Float64(x / fma(y, fma(b, Float64(b * 0.5), b), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -2.4e+76], N[(N[(x * N[(b * N[(b * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 3.6e+113], N[(x / N[(b * N[(b * N[(N[(y * a), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(b * N[(b * 0.5), $MachinePrecision] + b), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4e76

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 98.2

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 98.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 92.7

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 92.7%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{x + b \cdot \left(-1 \cdot x + \frac{1}{2} \cdot \left(b \cdot x\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{b \cdot \left(-1 \cdot x + \frac{1}{2} \cdot \left(b \cdot x\right)\right) + x}}{y} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\color{blue}{\left(b \cdot \left(-1 \cdot x\right) + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot x\right)\right)\right)} + x}{y} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\left(\color{blue}{\left(b \cdot -1\right) \cdot x} + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot x\right)\right)\right) + x}{y} \]

       4. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(-1 \cdot b\right)} \cdot x + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot x\right)\right)\right) + x}{y} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\left(\left(-1 \cdot b\right) \cdot x + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot x\right)}\right) + x}{y} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\left(\left(-1 \cdot b\right) \cdot x + \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot x}\right) + x}{y} \]

       7. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(-1 \cdot b + b \cdot \left(\frac{1}{2} \cdot b\right)\right)} + x}{y} \]

       8. *-commutative N/A

          \[\leadsto \frac{x \cdot \left(\color{blue}{b \cdot -1} + b \cdot \left(\frac{1}{2} \cdot b\right)\right) + x}{y} \]

       9. distribute-lft-in N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(-1 + \frac{1}{2} \cdot b\right)\right)} + x}{y} \]

       10. +-commutative N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + -1\right)}\right) + x}{y} \]

       11. metadata-eval N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \left(\frac{1}{2} \cdot b + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) + x}{y} \]

       12. sub-neg N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot b - 1\right)}\right) + x}{y} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, b \cdot \left(\frac{1}{2} \cdot b - 1\right), x\right)}}{y} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b - 1\right)}, x\right)}{y} \]

       15. sub-neg N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)\right)}, x\right)}{y} \]

       16. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \left(\color{blue}{b \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right)}{y} \]

       17. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \left(b \cdot \frac{1}{2} + \color{blue}{-1}\right), x\right)}{y} \]

       18. accelerator-lowering-fma.f64 82.0

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \color{blue}{\mathsf{fma}\left(b, 0.5, -1\right)}, x\right)}{y} \]

   11. Simplified 82.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, b \cdot \mathsf{fma}\left(b, 0.5, -1\right), x\right)}}{y} \]

   if -2.4e76 < b < 3.59999999999999992e113

   1. Initial program 98.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 65.4

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 65.4%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 42.6

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)} + y\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right)\right) + y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b\right)}\right) + y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right)}\right) + y\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)}\right)\right) + y\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot y\right)}\right) + y\right)} \]

       8. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot y}\right) + y\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot \left(b + b \cdot \left(\frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       10. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{b \cdot 1} + b \cdot \left(\frac{1}{2} \cdot b\right)\right) + y\right)} \]

       11. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b \cdot \left(1 + \frac{1}{2} \cdot b\right), y\right)}} \]

       13. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + 1\right)}, y\right)} \]

       14. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b\right) + b \cdot 1}, y\right)} \]

       15. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{b}, y\right)} \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot b, b\right)}, y\right)} \]

       17. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{2}}, b\right), y\right)} \]

       18. *-lowering-*.f64 33.9

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot 0.5}, b\right), y\right)} \]

   11. Simplified 33.9%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{\color{blue}{\frac{1}{2} \cdot \left(a \cdot \left({b}^{2} \cdot y\right)\right)}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{x}{\frac{1}{2} \cdot \left(a \cdot \color{blue}{\left(y \cdot {b}^{2}\right)}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{x}{\frac{1}{2} \cdot \color{blue}{\left(\left(a \cdot y\right) \cdot {b}^{2}\right)}} \]

       3. associate-*l* N/A

          \[\leadsto \frac{x}{\color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot y\right)\right) \cdot {b}^{2}}} \]

       4. unpow2 N/A

          \[\leadsto \frac{x}{\left(\frac{1}{2} \cdot \left(a \cdot y\right)\right) \cdot \color{blue}{\left(b \cdot b\right)}} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{\color{blue}{\left(\left(\frac{1}{2} \cdot \left(a \cdot y\right)\right) \cdot b\right) \cdot b}} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot \left(a \cdot y\right)\right) \cdot b\right)}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot \left(\left(\frac{1}{2} \cdot \left(a \cdot y\right)\right) \cdot b\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)}} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{b \cdot \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \left(a \cdot y\right)\right)\right)}} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot y\right)\right)}\right)} \]

       11. *-commutative N/A

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot a\right)}\right)\right)} \]

       12. *-lowering-*.f64 35.8

           \[\leadsto \frac{x}{b \cdot \left(b \cdot \left(0.5 \cdot \color{blue}{\left(y \cdot a\right)}\right)\right)} \]

   14. Simplified 35.8%

       \[\leadsto \frac{x}{\color{blue}{b \cdot \left(b \cdot \left(0.5 \cdot \left(y \cdot a\right)\right)\right)}} \]

   if 3.59999999999999992e113 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 95.0

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 95.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 89.9

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 89.9%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]
   9. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{e^{b}}}}{y} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{e^{b}}}}{y} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      6. exp-lowering-exp.f64 89.9

         \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 89.9%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{x}{\color{blue}{y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)}} \]
   12. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \frac{x}{y + \color{blue}{\left(y \cdot b + \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right) \cdot b\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{y + \left(\color{blue}{b \cdot y} + \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right) \cdot b\right)} \]

       3. associate-+r+ N/A

          \[\leadsto \frac{x}{\color{blue}{\left(y + b \cdot y\right) + \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right) \cdot b}} \]

       4. associate-*l* N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \color{blue}{\frac{1}{2} \cdot \left(\left(b \cdot y\right) \cdot b\right)}} \]

       5. *-commutative N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \frac{1}{2} \cdot \left(\color{blue}{\left(y \cdot b\right)} \cdot b\right)} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \frac{1}{2} \cdot \color{blue}{\left(y \cdot \left(b \cdot b\right)\right)}} \]

       7. unpow2 N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \frac{1}{2} \cdot \left(y \cdot \color{blue}{{b}^{2}}\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \frac{1}{2} \cdot \color{blue}{\left({b}^{2} \cdot y\right)}} \]

       9. associate-*r* N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right) \cdot y}} \]

       10. associate-+r+ N/A

           \[\leadsto \frac{x}{\color{blue}{y + \left(b \cdot y + \left(\frac{1}{2} \cdot {b}^{2}\right) \cdot y\right)}} \]

       11. distribute-rgt-in N/A

           \[\leadsto \frac{x}{y + \color{blue}{y \cdot \left(b + \frac{1}{2} \cdot {b}^{2}\right)}} \]

       12. +-commutative N/A

           \[\leadsto \frac{x}{\color{blue}{y \cdot \left(b + \frac{1}{2} \cdot {b}^{2}\right) + y}} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, b + \frac{1}{2} \cdot {b}^{2}, y\right)}} \]

       14. +-commutative N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot {b}^{2} + b}, y\right)} \]

       15. unpow2 N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)} + b, y\right)} \]

       16. associate-*r* N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\left(\frac{1}{2} \cdot b\right) \cdot b} + b, y\right)} \]

       17. *-commutative N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b\right)} + b, y\right)} \]

       18. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot b, b\right)}, y\right)} \]

       19. *-commutative N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{2}}, b\right), y\right)} \]

       20. *-lowering-*.f64 80.2

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot 0.5}, b\right), y\right)} \]

   13. Simplified 80.2%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, b \cdot \mathsf{fma}\left(b, 0.5, -1\right), x\right)}{y}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+113}:\\ \;\;\;\;\frac{x}{b \cdot \left(b \cdot \left(\left(y \cdot a\right) \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, b \cdot \mathsf{fma}\left(b, 0.5, -1\right), x\right)}{y}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -9.5e+60)
   (/ (fma x (* b (fma b 0.5 -1.0)) x) y)
   (if (<= b 2.7e+112) (/ (/ x a) y) (/ x (fma y (fma b (* b 0.5) b) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -9.5e+60) {
		tmp = fma(x, (b * fma(b, 0.5, -1.0)), x) / y;
	} else if (b <= 2.7e+112) {
		tmp = (x / a) / y;
	} else {
		tmp = x / fma(y, fma(b, (b * 0.5), b), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -9.5e+60)
		tmp = Float64(fma(x, Float64(b * fma(b, 0.5, -1.0)), x) / y);
	elseif (b <= 2.7e+112)
		tmp = Float64(Float64(x / a) / y);
	else
		tmp = Float64(x / fma(y, fma(b, Float64(b * 0.5), b), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -9.5e+60], N[(N[(x * N[(b * N[(b * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 2.7e+112], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(y * N[(b * N[(b * 0.5), $MachinePrecision] + b), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.49999999999999988e60

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 94.9

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 94.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 88.1

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 88.1%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{x + b \cdot \left(-1 \cdot x + \frac{1}{2} \cdot \left(b \cdot x\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{b \cdot \left(-1 \cdot x + \frac{1}{2} \cdot \left(b \cdot x\right)\right) + x}}{y} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\color{blue}{\left(b \cdot \left(-1 \cdot x\right) + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot x\right)\right)\right)} + x}{y} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\left(\color{blue}{\left(b \cdot -1\right) \cdot x} + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot x\right)\right)\right) + x}{y} \]

       4. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(-1 \cdot b\right)} \cdot x + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot x\right)\right)\right) + x}{y} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\left(\left(-1 \cdot b\right) \cdot x + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot x\right)}\right) + x}{y} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\left(\left(-1 \cdot b\right) \cdot x + \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot x}\right) + x}{y} \]

       7. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(-1 \cdot b + b \cdot \left(\frac{1}{2} \cdot b\right)\right)} + x}{y} \]

       8. *-commutative N/A

          \[\leadsto \frac{x \cdot \left(\color{blue}{b \cdot -1} + b \cdot \left(\frac{1}{2} \cdot b\right)\right) + x}{y} \]

       9. distribute-lft-in N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(-1 + \frac{1}{2} \cdot b\right)\right)} + x}{y} \]

       10. +-commutative N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + -1\right)}\right) + x}{y} \]

       11. metadata-eval N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \left(\frac{1}{2} \cdot b + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) + x}{y} \]

       12. sub-neg N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot b - 1\right)}\right) + x}{y} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, b \cdot \left(\frac{1}{2} \cdot b - 1\right), x\right)}}{y} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b - 1\right)}, x\right)}{y} \]

       15. sub-neg N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)\right)}, x\right)}{y} \]

       16. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \left(\color{blue}{b \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right)}{y} \]

       17. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \left(b \cdot \frac{1}{2} + \color{blue}{-1}\right), x\right)}{y} \]

       18. accelerator-lowering-fma.f64 78.2

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \color{blue}{\mathsf{fma}\left(b, 0.5, -1\right)}, x\right)}{y} \]

   11. Simplified 78.2%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, b \cdot \mathsf{fma}\left(b, 0.5, -1\right), x\right)}}{y} \]

   if -9.49999999999999988e60 < b < 2.7000000000000001e112

   1. Initial program 98.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 66.4

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 66.4%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Simplified 67.5%

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 32.4

          \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

   11. Simplified 32.4%

       \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

   if 2.7000000000000001e112 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 95.0

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 95.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 89.9

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 89.9%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]
   9. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{e^{b}}}}{y} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{e^{b}}}}{y} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      6. exp-lowering-exp.f64 89.9

         \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 89.9%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{x}{\color{blue}{y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)}} \]
   12. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \frac{x}{y + \color{blue}{\left(y \cdot b + \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right) \cdot b\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{y + \left(\color{blue}{b \cdot y} + \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right) \cdot b\right)} \]

       3. associate-+r+ N/A

          \[\leadsto \frac{x}{\color{blue}{\left(y + b \cdot y\right) + \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right) \cdot b}} \]

       4. associate-*l* N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \color{blue}{\frac{1}{2} \cdot \left(\left(b \cdot y\right) \cdot b\right)}} \]

       5. *-commutative N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \frac{1}{2} \cdot \left(\color{blue}{\left(y \cdot b\right)} \cdot b\right)} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \frac{1}{2} \cdot \color{blue}{\left(y \cdot \left(b \cdot b\right)\right)}} \]

       7. unpow2 N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \frac{1}{2} \cdot \left(y \cdot \color{blue}{{b}^{2}}\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \frac{1}{2} \cdot \color{blue}{\left({b}^{2} \cdot y\right)}} \]

       9. associate-*r* N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right) \cdot y}} \]

       10. associate-+r+ N/A

           \[\leadsto \frac{x}{\color{blue}{y + \left(b \cdot y + \left(\frac{1}{2} \cdot {b}^{2}\right) \cdot y\right)}} \]

       11. distribute-rgt-in N/A

           \[\leadsto \frac{x}{y + \color{blue}{y \cdot \left(b + \frac{1}{2} \cdot {b}^{2}\right)}} \]

       12. +-commutative N/A

           \[\leadsto \frac{x}{\color{blue}{y \cdot \left(b + \frac{1}{2} \cdot {b}^{2}\right) + y}} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, b + \frac{1}{2} \cdot {b}^{2}, y\right)}} \]

       14. +-commutative N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot {b}^{2} + b}, y\right)} \]

       15. unpow2 N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)} + b, y\right)} \]

       16. associate-*r* N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\left(\frac{1}{2} \cdot b\right) \cdot b} + b, y\right)} \]

       17. *-commutative N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b\right)} + b, y\right)} \]

       18. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot b, b\right)}, y\right)} \]

       19. *-commutative N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{2}}, b\right), y\right)} \]

       20. *-lowering-*.f64 80.2

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot 0.5}, b\right), y\right)} \]

   13. Simplified 80.2%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 50.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -2.8e+75)
   (/ (* x (fma b (fma b (fma b -0.16666666666666666 0.5) -1.0) 1.0)) y)
   (/ (* x 2.0) (* y (* a (* b b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.8e+75) {
		tmp = (x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y;
	} else {
		tmp = (x * 2.0) / (y * (a * (b * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -2.8e+75)
		tmp = Float64(Float64(x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y);
	else
		tmp = Float64(Float64(x * 2.0) / Float64(y * Float64(a * Float64(b * b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -2.8e+75], N[(N[(x * N[(b * N[(b * N[(b * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(y * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.80000000000000012e75

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 98.2

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 98.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 92.7

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 92.7%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right) + 1\right)}}{y} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1, 1\right)}}{y} \]

       3. sub-neg N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{y} \]

       4. metadata-eval N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \color{blue}{-1}, 1\right)}{y} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{-1}{6} \cdot b, -1\right)}, 1\right)}{y} \]

       6. +-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot b + \frac{1}{2}}, -1\right), 1\right)}{y} \]

       7. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{y} \]

       8. accelerator-lowering-fma.f64 89.1

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{y} \]

   11. Simplified 89.1%

       \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{y} \]

   if -2.80000000000000012e75 < b

   1. Initial program 99.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 67.1

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 51.8

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 51.8%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)} + y\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right)\right) + y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b\right)}\right) + y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right)}\right) + y\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)}\right)\right) + y\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot y\right)}\right) + y\right)} \]

       8. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot y}\right) + y\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot \left(b + b \cdot \left(\frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       10. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{b \cdot 1} + b \cdot \left(\frac{1}{2} \cdot b\right)\right) + y\right)} \]

       11. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b \cdot \left(1 + \frac{1}{2} \cdot b\right), y\right)}} \]

       13. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + 1\right)}, y\right)} \]

       14. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b\right) + b \cdot 1}, y\right)} \]

       15. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{b}, y\right)} \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot b, b\right)}, y\right)} \]

       17. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{2}}, b\right), y\right)} \]

       18. *-lowering-*.f64 42.8

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot 0.5}, b\right), y\right)} \]

   11. Simplified 42.8%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{a \cdot \left({b}^{2} \cdot y\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(a \cdot {b}^{2}\right) \cdot y}} \]

       6. *-commutative N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot \left(a \cdot {b}^{2}\right)}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot \left(a \cdot {b}^{2}\right)}} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{y \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{x \cdot 2}{y \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

       10. *-lowering-*.f64 47.6

           \[\leadsto \frac{x \cdot 2}{y \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

   14. Simplified 47.6%

       \[\leadsto \color{blue}{\frac{x \cdot 2}{y \cdot \left(a \cdot \left(b \cdot b\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 49.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, b \cdot \mathsf{fma}\left(b, 0.5, -1\right), x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot \left(a \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.02e+76)
   (/ (fma x (* b (fma b 0.5 -1.0)) x) y)
   (/ (* x 2.0) (* y (* a (* b b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.02e+76) {
		tmp = fma(x, (b * fma(b, 0.5, -1.0)), x) / y;
	} else {
		tmp = (x * 2.0) / (y * (a * (b * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.02e+76)
		tmp = Float64(fma(x, Float64(b * fma(b, 0.5, -1.0)), x) / y);
	else
		tmp = Float64(Float64(x * 2.0) / Float64(y * Float64(a * Float64(b * b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.02e+76], N[(N[(x * N[(b * N[(b * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(y * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.02000000000000007e76

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 98.2

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 98.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 92.7

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 92.7%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{x + b \cdot \left(-1 \cdot x + \frac{1}{2} \cdot \left(b \cdot x\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{b \cdot \left(-1 \cdot x + \frac{1}{2} \cdot \left(b \cdot x\right)\right) + x}}{y} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\color{blue}{\left(b \cdot \left(-1 \cdot x\right) + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot x\right)\right)\right)} + x}{y} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\left(\color{blue}{\left(b \cdot -1\right) \cdot x} + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot x\right)\right)\right) + x}{y} \]

       4. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(-1 \cdot b\right)} \cdot x + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot x\right)\right)\right) + x}{y} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\left(\left(-1 \cdot b\right) \cdot x + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot x\right)}\right) + x}{y} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\left(\left(-1 \cdot b\right) \cdot x + \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot x}\right) + x}{y} \]

       7. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(-1 \cdot b + b \cdot \left(\frac{1}{2} \cdot b\right)\right)} + x}{y} \]

       8. *-commutative N/A

          \[\leadsto \frac{x \cdot \left(\color{blue}{b \cdot -1} + b \cdot \left(\frac{1}{2} \cdot b\right)\right) + x}{y} \]

       9. distribute-lft-in N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(-1 + \frac{1}{2} \cdot b\right)\right)} + x}{y} \]

       10. +-commutative N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + -1\right)}\right) + x}{y} \]

       11. metadata-eval N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \left(\frac{1}{2} \cdot b + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) + x}{y} \]

       12. sub-neg N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot b - 1\right)}\right) + x}{y} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, b \cdot \left(\frac{1}{2} \cdot b - 1\right), x\right)}}{y} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b - 1\right)}, x\right)}{y} \]

       15. sub-neg N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)\right)}, x\right)}{y} \]

       16. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \left(\color{blue}{b \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right)}{y} \]

       17. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \left(b \cdot \frac{1}{2} + \color{blue}{-1}\right), x\right)}{y} \]

       18. accelerator-lowering-fma.f64 82.0

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \color{blue}{\mathsf{fma}\left(b, 0.5, -1\right)}, x\right)}{y} \]

   11. Simplified 82.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, b \cdot \mathsf{fma}\left(b, 0.5, -1\right), x\right)}}{y} \]

   if -1.02000000000000007e76 < b

   1. Initial program 99.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 67.1

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 51.8

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 51.8%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)} + y\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot b\right)}\right)\right) + y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot b\right)}\right) + y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(y \cdot b\right)\right)}\right) + y\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot y\right)}\right)\right) + y\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot y\right)}\right) + y\right)} \]

       8. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot y + \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot y}\right) + y\right)} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot \left(b + b \cdot \left(\frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       10. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{b \cdot 1} + b \cdot \left(\frac{1}{2} \cdot b\right)\right) + y\right)} \]

       11. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)} + y\right)} \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b \cdot \left(1 + \frac{1}{2} \cdot b\right), y\right)}} \]

       13. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + 1\right)}, y\right)} \]

       14. distribute-lft-in N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b\right) + b \cdot 1}, y\right)} \]

       15. *-rgt-identity N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, b \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{b}, y\right)} \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot b, b\right)}, y\right)} \]

       17. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{2}}, b\right), y\right)} \]

       18. *-lowering-*.f64 42.8

           \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot 0.5}, b\right), y\right)} \]

   11. Simplified 42.8%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{2 \cdot \frac{x}{a \cdot \left({b}^{2} \cdot y\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{a \cdot \left({b}^{2} \cdot y\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{a \cdot \left({b}^{2} \cdot y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(a \cdot {b}^{2}\right) \cdot y}} \]

       6. *-commutative N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot \left(a \cdot {b}^{2}\right)}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot \left(a \cdot {b}^{2}\right)}} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot 2}{y \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{x \cdot 2}{y \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

       10. *-lowering-*.f64 47.6

           \[\leadsto \frac{x \cdot 2}{y \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

   14. Simplified 47.6%

       \[\leadsto \color{blue}{\frac{x \cdot 2}{y \cdot \left(a \cdot \left(b \cdot b\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 38.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{x - x \cdot b}{y}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.2e+86)
   (/ (- x (* x b)) y)
   (if (<= b 1.45e+23) (/ (/ x a) y) (/ x (* y (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.2e+86) {
		tmp = (x - (x * b)) / y;
	} else if (b <= 1.45e+23) {
		tmp = (x / a) / y;
	} else {
		tmp = x / (y * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.2d+86)) then
        tmp = (x - (x * b)) / y
    else if (b <= 1.45d+23) then
        tmp = (x / a) / y
    else
        tmp = x / (y * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.2e+86) {
		tmp = (x - (x * b)) / y;
	} else if (b <= 1.45e+23) {
		tmp = (x / a) / y;
	} else {
		tmp = x / (y * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1.2e+86:
		tmp = (x - (x * b)) / y
	elif b <= 1.45e+23:
		tmp = (x / a) / y
	else:
		tmp = x / (y * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.2e+86)
		tmp = Float64(Float64(x - Float64(x * b)) / y);
	elseif (b <= 1.45e+23)
		tmp = Float64(Float64(x / a) / y);
	else
		tmp = Float64(x / Float64(y * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.2e+86)
		tmp = (x - (x * b)) / y;
	elseif (b <= 1.45e+23)
		tmp = (x / a) / y;
	else
		tmp = x / (y * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.2e+86], N[(N[(x - N[(x * b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 1.45e+23], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(y * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.2e86

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 98.1

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 98.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 92.6

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 92.6%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \left(b \cdot x\right)}}{y} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{x + \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)}}{y} \]

       2. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       3. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       4. *-commutative N/A

          \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y} \]

       5. *-lowering-*.f64 42.2

          \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y} \]

   11. Simplified 42.2%

       \[\leadsto \frac{\color{blue}{x - x \cdot b}}{y} \]

   if -1.2e86 < b < 1.45000000000000006e23

   1. Initial program 98.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 68.9

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 68.9%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Simplified 68.0%

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 35.0

          \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

   11. Simplified 35.0%

       \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

   if 1.45000000000000006e23 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 63.4

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 63.4%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 82.0

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot b} + y\right)} \]

       5. accelerator-lowering-fma.f64 37.9

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   11. Simplified 37.9%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(y, b, y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{x}{a \cdot \left(b \cdot y\right)}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot \left(b \cdot y\right)}} \]

       2. associate-*r* N/A

          \[\leadsto \frac{x}{\color{blue}{\left(a \cdot b\right) \cdot y}} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y \cdot \left(a \cdot b\right)}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y \cdot \left(a \cdot b\right)}} \]

       5. *-commutative N/A

          \[\leadsto \frac{x}{y \cdot \color{blue}{\left(b \cdot a\right)}} \]

       6. *-lowering-*.f64 42.8

          \[\leadsto \frac{x}{y \cdot \color{blue}{\left(b \cdot a\right)}} \]

   14. Simplified 42.8%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \left(b \cdot a\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{x - x \cdot b}{y}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 38000000:\\ \;\;\;\;\left(1 - b\right) \cdot \frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 38000000.0)
   (* (- 1.0 b) (/ x (* y a)))
   (/ x (fma y (fma b (* b 0.5) b) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 38000000.0) {
		tmp = (1.0 - b) * (x / (y * a));
	} else {
		tmp = x / fma(y, fma(b, (b * 0.5), b), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 38000000.0)
		tmp = Float64(Float64(1.0 - b) * Float64(x / Float64(y * a)));
	else
		tmp = Float64(x / fma(y, fma(b, Float64(b * 0.5), b), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 38000000.0], N[(N[(1.0 - b), $MachinePrecision] * N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(b * N[(b * 0.5), $MachinePrecision] + b), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8e7

   1. Initial program 98.9%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 70.8

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.8%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 53.9

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 53.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \frac{x}{a \cdot y}\right)} + \frac{x}{a \cdot y} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot b\right) \cdot \frac{x}{a \cdot y}} + \frac{x}{a \cdot y} \]

       3. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(-1 \cdot b + 1\right) \cdot \frac{x}{a \cdot y}} \]

       4. +-commutative N/A

          \[\leadsto \color{blue}{\left(1 + -1 \cdot b\right)} \cdot \frac{x}{a \cdot y} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + -1 \cdot b\right) \cdot \frac{x}{a \cdot y}} \]

       6. neg-mul-1 N/A

          \[\leadsto \left(1 + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \cdot \frac{x}{a \cdot y} \]

       7. unsub-neg N/A

          \[\leadsto \color{blue}{\left(1 - b\right)} \cdot \frac{x}{a \cdot y} \]

       8. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 - b\right)} \cdot \frac{x}{a \cdot y} \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \left(1 - b\right) \cdot \color{blue}{\frac{x}{a \cdot y}} \]

       10. *-lowering-*.f64 35.8

           \[\leadsto \left(1 - b\right) \cdot \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 35.8%

       \[\leadsto \color{blue}{\left(1 - b\right) \cdot \frac{x}{a \cdot y}} \]

   if 3.8e7 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 89.6

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 89.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 79.1

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 79.1%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]
   9. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{e^{b}}}}{y} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{e^{b}}}}{y} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      6. exp-lowering-exp.f64 79.1

         \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 79.1%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{x}{\color{blue}{y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)}} \]
   12. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \frac{x}{y + \color{blue}{\left(y \cdot b + \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right) \cdot b\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{y + \left(\color{blue}{b \cdot y} + \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right) \cdot b\right)} \]

       3. associate-+r+ N/A

          \[\leadsto \frac{x}{\color{blue}{\left(y + b \cdot y\right) + \left(\frac{1}{2} \cdot \left(b \cdot y\right)\right) \cdot b}} \]

       4. associate-*l* N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \color{blue}{\frac{1}{2} \cdot \left(\left(b \cdot y\right) \cdot b\right)}} \]

       5. *-commutative N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \frac{1}{2} \cdot \left(\color{blue}{\left(y \cdot b\right)} \cdot b\right)} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \frac{1}{2} \cdot \color{blue}{\left(y \cdot \left(b \cdot b\right)\right)}} \]

       7. unpow2 N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \frac{1}{2} \cdot \left(y \cdot \color{blue}{{b}^{2}}\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \frac{1}{2} \cdot \color{blue}{\left({b}^{2} \cdot y\right)}} \]

       9. associate-*r* N/A

          \[\leadsto \frac{x}{\left(y + b \cdot y\right) + \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right) \cdot y}} \]

       10. associate-+r+ N/A

           \[\leadsto \frac{x}{\color{blue}{y + \left(b \cdot y + \left(\frac{1}{2} \cdot {b}^{2}\right) \cdot y\right)}} \]

       11. distribute-rgt-in N/A

           \[\leadsto \frac{x}{y + \color{blue}{y \cdot \left(b + \frac{1}{2} \cdot {b}^{2}\right)}} \]

       12. +-commutative N/A

           \[\leadsto \frac{x}{\color{blue}{y \cdot \left(b + \frac{1}{2} \cdot {b}^{2}\right) + y}} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, b + \frac{1}{2} \cdot {b}^{2}, y\right)}} \]

       14. +-commutative N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot {b}^{2} + b}, y\right)} \]

       15. unpow2 N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)} + b, y\right)} \]

       16. associate-*r* N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\left(\frac{1}{2} \cdot b\right) \cdot b} + b, y\right)} \]

       17. *-commutative N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b\right)} + b, y\right)} \]

       18. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot b, b\right)}, y\right)} \]

       19. *-commutative N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{2}}, b\right), y\right)} \]

       20. *-lowering-*.f64 53.1

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \color{blue}{b \cdot 0.5}, b\right), y\right)} \]

   13. Simplified 53.1%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 38000000:\\ \;\;\;\;\left(1 - b\right) \cdot \frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, b \cdot 0.5, b\right), y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 38.9% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+103}:\\ \;\;\;\;\frac{x - x \cdot b}{y}\\ \mathbf{elif}\;b \leq 2950:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.35e+103)
   (/ (- x (* x b)) y)
   (if (<= b 2950.0) (/ x (* y a)) (/ x (* y (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.35e+103) {
		tmp = (x - (x * b)) / y;
	} else if (b <= 2950.0) {
		tmp = x / (y * a);
	} else {
		tmp = x / (y * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.35d+103)) then
        tmp = (x - (x * b)) / y
    else if (b <= 2950.0d0) then
        tmp = x / (y * a)
    else
        tmp = x / (y * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.35e+103) {
		tmp = (x - (x * b)) / y;
	} else if (b <= 2950.0) {
		tmp = x / (y * a);
	} else {
		tmp = x / (y * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1.35e+103:
		tmp = (x - (x * b)) / y
	elif b <= 2950.0:
		tmp = x / (y * a)
	else:
		tmp = x / (y * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.35e+103)
		tmp = Float64(Float64(x - Float64(x * b)) / y);
	elseif (b <= 2950.0)
		tmp = Float64(x / Float64(y * a));
	else
		tmp = Float64(x / Float64(y * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.35e+103)
		tmp = (x - (x * b)) / y;
	elseif (b <= 2950.0)
		tmp = x / (y * a);
	else
		tmp = x / (y * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.35e+103], N[(N[(x - N[(x * b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 2950.0], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.34999999999999996e103

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 97.9

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 97.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 91.8

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 91.8%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \left(b \cdot x\right)}}{y} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{x + \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)}}{y} \]

       2. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       3. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       4. *-commutative N/A

          \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y} \]

       5. *-lowering-*.f64 44.3

          \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y} \]

   11. Simplified 44.3%

       \[\leadsto \frac{\color{blue}{x - x \cdot b}}{y} \]

   if -1.34999999999999996e103 < b < 2950

   1. Initial program 98.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 70.1

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 39.8

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 39.8%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

       2. *-lowering-*.f64 33.4

          \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 33.4%

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

   if 2950 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 63.9

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 80.0

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 80.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot b} + y\right)} \]

       5. accelerator-lowering-fma.f64 34.8

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   11. Simplified 34.8%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(y, b, y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{x}{a \cdot \left(b \cdot y\right)}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot \left(b \cdot y\right)}} \]

       2. associate-*r* N/A

          \[\leadsto \frac{x}{\color{blue}{\left(a \cdot b\right) \cdot y}} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y \cdot \left(a \cdot b\right)}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y \cdot \left(a \cdot b\right)}} \]

       5. *-commutative N/A

          \[\leadsto \frac{x}{y \cdot \color{blue}{\left(b \cdot a\right)}} \]

       6. *-lowering-*.f64 39.1

          \[\leadsto \frac{x}{y \cdot \color{blue}{\left(b \cdot a\right)}} \]

   14. Simplified 39.1%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \left(b \cdot a\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+103}:\\ \;\;\;\;\frac{x - x \cdot b}{y}\\ \mathbf{elif}\;b \leq 2950:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 38.5% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+103}:\\ \;\;\;\;\frac{x - x \cdot b}{y}\\ \mathbf{elif}\;b \leq 5700:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.7e+103)
   (/ (- x (* x b)) y)
   (if (<= b 5700.0) (/ x (* y a)) (/ x (* a (* y b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.7e+103) {
		tmp = (x - (x * b)) / y;
	} else if (b <= 5700.0) {
		tmp = x / (y * a);
	} else {
		tmp = x / (a * (y * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.7d+103)) then
        tmp = (x - (x * b)) / y
    else if (b <= 5700.0d0) then
        tmp = x / (y * a)
    else
        tmp = x / (a * (y * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.7e+103) {
		tmp = (x - (x * b)) / y;
	} else if (b <= 5700.0) {
		tmp = x / (y * a);
	} else {
		tmp = x / (a * (y * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1.7e+103:
		tmp = (x - (x * b)) / y
	elif b <= 5700.0:
		tmp = x / (y * a)
	else:
		tmp = x / (a * (y * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.7e+103)
		tmp = Float64(Float64(x - Float64(x * b)) / y);
	elseif (b <= 5700.0)
		tmp = Float64(x / Float64(y * a));
	else
		tmp = Float64(x / Float64(a * Float64(y * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.7e+103)
		tmp = (x - (x * b)) / y;
	elseif (b <= 5700.0)
		tmp = x / (y * a);
	else
		tmp = x / (a * (y * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.7e+103], N[(N[(x - N[(x * b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 5700.0], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.6999999999999999e103

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 97.9

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 97.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 91.8

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 91.8%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \left(b \cdot x\right)}}{y} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{x + \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)}}{y} \]

       2. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       3. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       4. *-commutative N/A

          \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y} \]

       5. *-lowering-*.f64 44.3

          \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y} \]

   11. Simplified 44.3%

       \[\leadsto \frac{\color{blue}{x - x \cdot b}}{y} \]

   if -1.6999999999999999e103 < b < 5700

   1. Initial program 98.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 70.1

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 39.8

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 39.8%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

       2. *-lowering-*.f64 33.4

          \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 33.4%

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

   if 5700 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 63.9

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 80.0

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 80.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot b} + y\right)} \]

       5. accelerator-lowering-fma.f64 34.8

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   11. Simplified 34.8%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(y, b, y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y\right)}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot b\right)}} \]

       2. *-lowering-*.f64 34.8

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot b\right)}} \]

   14. Simplified 34.8%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot b\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+103}:\\ \;\;\;\;\frac{x - x \cdot b}{y}\\ \mathbf{elif}\;b \leq 5700:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 37.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+103}:\\ \;\;\;\;\frac{x - x \cdot b}{y}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, b, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.35e+103)
   (/ (- x (* x b)) y)
   (if (<= b 8.5e+90) (/ x (* y a)) (/ x (fma y b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.35e+103) {
		tmp = (x - (x * b)) / y;
	} else if (b <= 8.5e+90) {
		tmp = x / (y * a);
	} else {
		tmp = x / fma(y, b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.35e+103)
		tmp = Float64(Float64(x - Float64(x * b)) / y);
	elseif (b <= 8.5e+90)
		tmp = Float64(x / Float64(y * a));
	else
		tmp = Float64(x / fma(y, b, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.35e+103], N[(N[(x - N[(x * b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 8.5e+90], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * b + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.34999999999999996e103

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 97.9

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 97.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 91.8

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 91.8%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \left(b \cdot x\right)}}{y} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{x + \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)}}{y} \]

       2. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       3. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       4. *-commutative N/A

          \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y} \]

       5. *-lowering-*.f64 44.3

          \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y} \]

   11. Simplified 44.3%

       \[\leadsto \frac{\color{blue}{x - x \cdot b}}{y} \]

   if -1.34999999999999996e103 < b < 8.5000000000000002e90

   1. Initial program 98.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 66.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 66.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 44.3

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 44.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

       2. *-lowering-*.f64 30.2

          \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 30.2%

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

   if 8.5000000000000002e90 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 93.0

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 93.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 88.3

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 88.3%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]
   9. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{e^{b}}}}{y} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{e^{b}}}}{y} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      6. exp-lowering-exp.f64 88.3

         \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 88.3%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{x}{\color{blue}{y + b \cdot y}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot y + y}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y \cdot b} + y} \]

       3. accelerator-lowering-fma.f64 31.6

          \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   13. Simplified 31.6%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+103}:\\ \;\;\;\;\frac{x - x \cdot b}{y}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, b, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 36.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - b\right)\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, b, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.4e+103)
   (* (/ x y) (- 1.0 b))
   (if (<= b 4.6e+90) (/ x (* y a)) (/ x (fma y b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.4e+103) {
		tmp = (x / y) * (1.0 - b);
	} else if (b <= 4.6e+90) {
		tmp = x / (y * a);
	} else {
		tmp = x / fma(y, b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.4e+103)
		tmp = Float64(Float64(x / y) * Float64(1.0 - b));
	elseif (b <= 4.6e+90)
		tmp = Float64(x / Float64(y * a));
	else
		tmp = Float64(x / fma(y, b, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.4e+103], N[(N[(x / y), $MachinePrecision] * N[(1.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.6e+90], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * b + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.40000000000000004e103

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 97.9

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 97.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 91.8

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 91.8%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{y} + \frac{x}{y}} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \frac{x}{y}\right)} + \frac{x}{y} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot b\right) \cdot \frac{x}{y}} + \frac{x}{y} \]

       3. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(-1 \cdot b + 1\right) \cdot \frac{x}{y}} \]

       4. +-commutative N/A

          \[\leadsto \color{blue}{\left(1 + -1 \cdot b\right)} \cdot \frac{x}{y} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + -1 \cdot b\right) \cdot \frac{x}{y}} \]

       6. neg-mul-1 N/A

          \[\leadsto \left(1 + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \cdot \frac{x}{y} \]

       7. unsub-neg N/A

          \[\leadsto \color{blue}{\left(1 - b\right)} \cdot \frac{x}{y} \]

       8. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 - b\right)} \cdot \frac{x}{y} \]

       9. /-lowering-/.f64 36.4

          \[\leadsto \left(1 - b\right) \cdot \color{blue}{\frac{x}{y}} \]

   11. Simplified 36.4%

       \[\leadsto \color{blue}{\left(1 - b\right) \cdot \frac{x}{y}} \]

   if -1.40000000000000004e103 < b < 4.6e90

   1. Initial program 98.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 66.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 66.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 44.3

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 44.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

       2. *-lowering-*.f64 30.2

          \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 30.2%

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

   if 4.6e90 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 93.0

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 93.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 88.3

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 88.3%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]
   9. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{e^{b}}}}{y} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{e^{b}}}}{y} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      6. exp-lowering-exp.f64 88.3

         \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 88.3%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{x}{\color{blue}{y + b \cdot y}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot y + y}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y \cdot b} + y} \]

       3. accelerator-lowering-fma.f64 31.6

          \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   13. Simplified 31.6%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - b\right)\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, b, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 37.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{x - x \cdot b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -2.5e+76) (/ (- x (* x b)) y) (/ x (* a (fma y b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.5e+76) {
		tmp = (x - (x * b)) / y;
	} else {
		tmp = x / (a * fma(y, b, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -2.5e+76)
		tmp = Float64(Float64(x - Float64(x * b)) / y);
	else
		tmp = Float64(x / Float64(a * fma(y, b, y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -2.5e+76], N[(N[(x - N[(x * b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(a * N[(y * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.49999999999999996e76

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 98.2

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 98.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 92.7

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 92.7%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \left(b \cdot x\right)}}{y} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{x + \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)}}{y} \]

       2. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       3. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       4. *-commutative N/A

          \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y} \]

       5. *-lowering-*.f64 41.5

          \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y} \]

   11. Simplified 41.5%

       \[\leadsto \frac{\color{blue}{x - x \cdot b}}{y} \]

   if -2.49999999999999996e76 < b

   1. Initial program 99.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 67.1

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 51.8

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 51.8%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y \cdot b} + y\right)} \]

       5. accelerator-lowering-fma.f64 34.0

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   11. Simplified 34.0%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(y, b, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 34.5% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+96}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, b, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 2e+96) (/ x (* y a)) (/ x (fma y b y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 2e+96) {
		tmp = x / (y * a);
	} else {
		tmp = x / fma(y, b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 2e+96)
		tmp = Float64(x / Float64(y * a));
	else
		tmp = Float64(x / fma(y, b, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 2e+96], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * b + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.0000000000000001e96

   1. Initial program 99.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 67.5

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 54.9

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 54.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

       2. *-lowering-*.f64 27.8

          \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 27.8%

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

   if 2.0000000000000001e96 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 93.0

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 93.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 88.3

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 88.3%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]
   9. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{e^{b}}}}{y} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{e^{b}}}}{y} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      6. exp-lowering-exp.f64 88.3

         \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 88.3%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{x}{\color{blue}{y + b \cdot y}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{b \cdot y + y}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y \cdot b} + y} \]

       3. accelerator-lowering-fma.f64 31.6

          \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   13. Simplified 31.6%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+96}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, b, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 31.1% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot a} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ x (* y a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / (y * a);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x / (y * a)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x / (y * a);
}

Python

def code(x, y, z, t, a, b):
	return x / (y * a)

Julia

function code(x, y, z, t, a, b)
	return Float64(x / Float64(y * a))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x / (y * a);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

   2. exp-diff N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

   7. exp-prod N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

   8. pow-lowering-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

   9. rem-exp-log N/A

      \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

   10. sub-neg N/A

       \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

   14. exp-lowering-exp.f64 68.6

       \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

5. Simplified 68.6%

   \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

   4. exp-lowering-exp.f64 60.4

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

8. Simplified 60.4%

   \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

9. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

    2. *-lowering-*.f64 25.3

       \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

11. Simplified 25.3%

    \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

12. Final simplification 25.3%

    \[\leadsto \frac{x}{y \cdot a} \]
13. Add Preprocessing

Alternative 27: 15.7% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ x y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x / y;
}

Python

def code(x, y, z, t, a, b):
	return x / y

Julia

function code(x, y, z, t, a, b)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   3. rem-exp-log N/A

      \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

   4. log-lowering-log.f64 N/A

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

   5. rem-exp-log 74.0

      \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

5. Simplified 74.0%

   \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{x \cdot {a}^{t}}{y}} \]
7. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{{a}^{t}}{y}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \frac{{a}^{t}}{y}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto x \cdot \color{blue}{\frac{{a}^{t}}{y}} \]

   4. pow-lowering-pow.f64 47.5

      \[\leadsto x \cdot \frac{\color{blue}{{a}^{t}}}{y} \]

8. Simplified 47.5%

   \[\leadsto \color{blue}{x \cdot \frac{{a}^{t}}{y}} \]

9. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{x}{y}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 14.3

       \[\leadsto \color{blue}{\frac{x}{y}} \]

11. Simplified 14.3%

    \[\leadsto \color{blue}{\frac{x}{y}} \]
12. Add Preprocessing

Developer Target 1: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {a}^{\left(t - 1\right)}\\ t_2 := \frac{x \cdot \frac{t\_1}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot t\_1}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (pow a (- t 1.0)))
        (t_2 (/ (* x (/ t_1 y)) (- (+ b 1.0) (* y (log z))))))
   (if (< t -0.8845848504127471)
     t_2
     (if (< t 852031.2288374073)
       (/ (* (/ x y) t_1) (exp (- b (* (log z) y))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a ** (t - 1.0d0)
    t_2 = (x * (t_1 / y)) / ((b + 1.0d0) - (y * log(z)))
    if (t < (-0.8845848504127471d0)) then
        tmp = t_2
    else if (t < 852031.2288374073d0) then
        tmp = ((x / y) * t_1) / exp((b - (log(z) * y)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * Math.log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / Math.exp((b - (Math.log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(a, (t - 1.0))
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * math.log(z)))
	tmp = 0
	if t < -0.8845848504127471:
		tmp = t_2
	elif t < 852031.2288374073:
		tmp = ((x / y) * t_1) / math.exp((b - (math.log(z) * y)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = a ^ Float64(t - 1.0)
	t_2 = Float64(Float64(x * Float64(t_1 / y)) / Float64(Float64(b + 1.0) - Float64(y * log(z))))
	tmp = 0.0
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = Float64(Float64(Float64(x / y) * t_1) / exp(Float64(b - Float64(log(z) * y))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a ^ (t - 1.0);
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	tmp = 0.0;
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] / N[(N[(b + 1.0), $MachinePrecision] - N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -0.8845848504127471], t$95$2, If[Less[t, 852031.2288374073], N[(N[(N[(x / y), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Exp[N[(b - N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024204 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -8845848504127471/10000000000000000) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 8520312288374073/10000000000) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))