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

Percentage Accurate: 98.5% → 98.5%

Time: 27.6s

Alternatives: 27

Speedup: 1.0×

• 20.9% 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: 98.5% 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: 98.5% 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 98.9%

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

2. Final simplification 98.9%

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

Alternative 2: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+49} \lor \neg \left(y \leq 3.05 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(t + -1\right) \cdot \log a - b}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -7.5e+49) (not (<= y 3.05e-12)))
   (/ (* x (exp (- (- (* y (log z)) (log a)) b))) y)
   (/ x (/ y (exp (- (* (+ t -1.0) (log a)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7.5e+49) || !(y <= 3.05e-12)) {
		tmp = (x * exp((((y * log(z)) - log(a)) - b))) / y;
	} else {
		tmp = x / (y / exp((((t + -1.0) * log(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 ((y <= (-7.5d+49)) .or. (.not. (y <= 3.05d-12))) then
        tmp = (x * exp((((y * log(z)) - log(a)) - b))) / y
    else
        tmp = x / (y / exp((((t + (-1.0d0)) * log(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 ((y <= -7.5e+49) || !(y <= 3.05e-12)) {
		tmp = (x * Math.exp((((y * Math.log(z)) - Math.log(a)) - b))) / y;
	} else {
		tmp = x / (y / Math.exp((((t + -1.0) * Math.log(a)) - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -7.5e+49) or not (y <= 3.05e-12):
		tmp = (x * math.exp((((y * math.log(z)) - math.log(a)) - b))) / y
	else:
		tmp = x / (y / math.exp((((t + -1.0) * math.log(a)) - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -7.5e+49) || !(y <= 3.05e-12))
		tmp = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) - log(a)) - b))) / y);
	else
		tmp = Float64(x / Float64(y / exp(Float64(Float64(Float64(t + -1.0) * log(a)) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -7.5e+49) || ~((y <= 3.05e-12)))
		tmp = (x * exp((((y * log(z)) - log(a)) - b))) / y;
	else
		tmp = x / (y / exp((((t + -1.0) * log(a)) - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -7.5e+49], N[Not[LessEqual[y, 3.05e-12]], $MachinePrecision]], N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(y / N[Exp[N[(N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.4999999999999995e49 or 3.0500000000000001e-12 < y

   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. Step-by-step derivation

      1. associate-/l* 99.8%

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

      2. fma-def 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 94.3%

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

   if -7.4999999999999995e49 < y < 3.0500000000000001e-12

   1. Initial program 98.0%

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

      1. associate-/l* 97.3%

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

      2. fma-def 97.3%

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

      3. sub-neg 97.3%

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

      4. metadata-eval 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in y around 0 97.3%

      \[\leadsto \frac{x}{\frac{y}{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+49} \lor \neg \left(y \leq 3.05 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(t + -1\right) \cdot \log a - b}}}\\ \end{array} \]

Alternative 3: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t + -1 \leq -1 \cdot 10^{+17} \lor \neg \left(t + -1 \leq 2 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (+ t -1.0) -1e+17) (not (<= (+ t -1.0) 2e+133)))
   (/ (* x (pow a (+ t -1.0))) y)
   (* (/ (pow z y) a) (/ x (* y (exp b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((t + -1.0) <= -1e+17) || !((t + -1.0) <= 2e+133)) {
		tmp = (x * pow(a, (t + -1.0))) / y;
	} else {
		tmp = (pow(z, y) / a) * (x / (y * exp(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 (((t + (-1.0d0)) <= (-1d+17)) .or. (.not. ((t + (-1.0d0)) <= 2d+133))) then
        tmp = (x * (a ** (t + (-1.0d0)))) / y
    else
        tmp = ((z ** y) / a) * (x / (y * exp(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 (((t + -1.0) <= -1e+17) || !((t + -1.0) <= 2e+133)) {
		tmp = (x * Math.pow(a, (t + -1.0))) / y;
	} else {
		tmp = (Math.pow(z, y) / a) * (x / (y * Math.exp(b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((t + -1.0) <= -1e+17) or not ((t + -1.0) <= 2e+133):
		tmp = (x * math.pow(a, (t + -1.0))) / y
	else:
		tmp = (math.pow(z, y) / a) * (x / (y * math.exp(b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(t + -1.0) <= -1e+17) || !(Float64(t + -1.0) <= 2e+133))
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	else
		tmp = Float64(Float64((z ^ y) / a) * Float64(x / Float64(y * exp(b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((t + -1.0) <= -1e+17) || ~(((t + -1.0) <= 2e+133)))
		tmp = (x * (a ^ (t + -1.0))) / y;
	else
		tmp = ((z ^ y) / a) * (x / (y * exp(b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(t + -1.0), $MachinePrecision], -1e+17], N[Not[LessEqual[N[(t + -1.0), $MachinePrecision], 2e+133]], $MachinePrecision]], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] * N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 t 1) < -1e17 or 2e133 < (-.f64 t 1)

   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. Taylor expanded in y around 0 92.3%

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

      1. div-exp 73.1%

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

      2. exp-to-pow 73.1%

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

      3. sub-neg 73.1%

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

      4. metadata-eval 73.1%

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

   4. Simplified 73.1%

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

   5. Taylor expanded in b around 0 85.6%

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

      1. exp-to-pow 85.6%

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

      2. sub-neg 85.6%

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

      3. metadata-eval 85.6%

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

      4. +-commutative 85.6%

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

   7. Simplified 85.6%

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

   if -1e17 < (-.f64 t 1) < 2e133

   1. Initial program 98.3%

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

      1. associate-*l/ 89.5%

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

      2. *-commutative 89.5%

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

      3. +-commutative 89.5%

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

      4. associate--l+ 89.5%

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

      5. exp-sum 83.5%

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

      6. *-commutative 83.5%

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

      7. exp-to-pow 84.3%

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

      8. sub-neg 84.3%

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

      9. metadata-eval 84.3%

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

      10. exp-diff 75.3%

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

      11. *-commutative 75.3%

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

      12. exp-to-pow 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in t around 0 76.1%

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

      1. *-commutative 76.1%

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

      2. times-frac 80.0%

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

   6. Simplified 80.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t + -1 \leq -1 \cdot 10^{+17} \lor \neg \left(t + -1 \leq 2 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}\\ \end{array} \]

Alternative 4: 89.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+152} \lor \neg \left(y \leq 6 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(t + -1\right) \cdot \log a - b}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6.8e+152) (not (<= y 6e+38)))
   (/ (* x (/ (pow z y) a)) y)
   (/ x (/ y (exp (- (* (+ t -1.0) (log a)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.8e+152) || !(y <= 6e+38)) {
		tmp = (x * (pow(z, y) / a)) / y;
	} else {
		tmp = x / (y / exp((((t + -1.0) * log(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 ((y <= (-6.8d+152)) .or. (.not. (y <= 6d+38))) then
        tmp = (x * ((z ** y) / a)) / y
    else
        tmp = x / (y / exp((((t + (-1.0d0)) * log(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 ((y <= -6.8e+152) || !(y <= 6e+38)) {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	} else {
		tmp = x / (y / Math.exp((((t + -1.0) * Math.log(a)) - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6.8e+152) or not (y <= 6e+38):
		tmp = (x * (math.pow(z, y) / a)) / y
	else:
		tmp = x / (y / math.exp((((t + -1.0) * math.log(a)) - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.8e+152) || !(y <= 6e+38))
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	else
		tmp = Float64(x / Float64(y / exp(Float64(Float64(Float64(t + -1.0) * log(a)) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6.8e+152) || ~((y <= 6e+38)))
		tmp = (x * ((z ^ y) / a)) / y;
	else
		tmp = x / (y / exp((((t + -1.0) * log(a)) - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6.8e+152], N[Not[LessEqual[y, 6e+38]], $MachinePrecision]], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(y / N[Exp[N[(N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.80000000000000041e152 or 6.0000000000000002e38 < 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. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. fma-def 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 95.8%

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

   5. Taylor expanded in b around 0 92.6%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}}{y} \]
   6. Step-by-step derivation

      1. +-commutative 92.6%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 92.6%

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

      3. sub-neg 92.6%

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

      4. exp-diff 92.6%

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

      5. *-commutative 92.6%

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

      6. exp-to-pow 92.6%

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

      7. rem-exp-log 92.6%

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

      8. *-commutative 92.6%

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

   7. Simplified 92.6%

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

   if -6.80000000000000041e152 < y < 6.0000000000000002e38

   1. Initial program 98.2%

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

      1. associate-/l* 97.6%

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

      2. fma-def 97.6%

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

      3. sub-neg 97.6%

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

      4. metadata-eval 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in y around 0 94.1%

      \[\leadsto \frac{x}{\frac{y}{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+152} \lor \neg \left(y \leq 6 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(t + -1\right) \cdot \log a - b}}}\\ \end{array} \]

Alternative 5: 81.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.15 \cdot 10^{+152}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{\frac{y \cdot e^{b}}{{a}^{\left(t + -1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.15e+152)
   (/ (* x (/ (pow z y) a)) y)
   (if (<= y 3.7e-12)
     (/ x (/ (* y (exp b)) (pow a (+ t -1.0))))
     (/ (/ (* x (pow z y)) a) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.15e+152) {
		tmp = (x * (pow(z, y) / a)) / y;
	} else if (y <= 3.7e-12) {
		tmp = x / ((y * exp(b)) / pow(a, (t + -1.0)));
	} else {
		tmp = ((x * pow(z, y)) / a) / y;
	}
	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 (y <= (-4.15d+152)) then
        tmp = (x * ((z ** y) / a)) / y
    else if (y <= 3.7d-12) then
        tmp = x / ((y * exp(b)) / (a ** (t + (-1.0d0))))
    else
        tmp = ((x * (z ** y)) / a) / y
    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 (y <= -4.15e+152) {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	} else if (y <= 3.7e-12) {
		tmp = x / ((y * Math.exp(b)) / Math.pow(a, (t + -1.0)));
	} else {
		tmp = ((x * Math.pow(z, y)) / a) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -4.15e+152:
		tmp = (x * (math.pow(z, y) / a)) / y
	elif y <= 3.7e-12:
		tmp = x / ((y * math.exp(b)) / math.pow(a, (t + -1.0)))
	else:
		tmp = ((x * math.pow(z, y)) / a) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.15e+152)
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	elseif (y <= 3.7e-12)
		tmp = Float64(x / Float64(Float64(y * exp(b)) / (a ^ Float64(t + -1.0))));
	else
		tmp = Float64(Float64(Float64(x * (z ^ y)) / a) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -4.15e+152)
		tmp = (x * ((z ^ y) / a)) / y;
	elseif (y <= 3.7e-12)
		tmp = x / ((y * exp(b)) / (a ^ (t + -1.0)));
	else
		tmp = ((x * (z ^ y)) / a) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.15e+152], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 3.7e-12], N[(x / N[(N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision] / N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1500000000000001e152

   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. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. fma-def 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 95.1%

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

   5. Taylor expanded in b around 0 95.1%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}}{y} \]
   6. Step-by-step derivation

      1. +-commutative 95.1%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 95.1%

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

      3. sub-neg 95.1%

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

      4. exp-diff 95.1%

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

      5. *-commutative 95.1%

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

      6. exp-to-pow 95.1%

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

      7. rem-exp-log 95.1%

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

      8. *-commutative 95.1%

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

   7. Simplified 95.1%

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

   if -4.1500000000000001e152 < y < 3.69999999999999999e-12

   1. Initial program 98.3%

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

      1. associate-*l/ 94.7%

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

      2. *-commutative 94.7%

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

      3. +-commutative 94.7%

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

      4. associate--l+ 94.7%

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

      5. exp-sum 82.6%

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

      6. *-commutative 82.6%

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

      7. exp-to-pow 83.3%

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

      8. sub-neg 83.3%

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

      9. metadata-eval 83.3%

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

      10. exp-diff 79.4%

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

      11. *-commutative 79.4%

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

      12. exp-to-pow 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in y around 0 80.6%

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

      1. associate-/l* 85.7%

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

      2. exp-to-pow 86.2%

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

      3. sub-neg 86.2%

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

      4. metadata-eval 86.2%

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

   6. Simplified 86.2%

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

   if 3.69999999999999999e-12 < y

   1. Initial program 99.6%

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

      1. expm1-log1p-u 84.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right)\right)}}{y} \]

      2. expm1-udef 81.6%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right)} - 1}}{y} \]

   3. Applied egg-rr 46.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(\frac{{a}^{t}}{a} \cdot \frac{{z}^{y}}{e^{b}}\right)\right)} - 1}}{y} \]
   4. Step-by-step derivation

      1. expm1-def 49.1%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\frac{{a}^{t}}{a} \cdot \frac{{z}^{y}}{e^{b}}\right)\right)\right)}}{y} \]

      2. expm1-log1p 61.0%

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

      3. *-commutative 61.0%

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

      4. associate-*l/ 61.0%

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

      5. associate-*r/ 61.0%

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

      6. associate-/l/ 61.0%

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

      7. *-commutative 61.0%

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

   5. Simplified 61.0%

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

   6. Taylor expanded in t around 0 74.6%

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

      1. associate-/r* 74.6%

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

   8. Simplified 74.6%

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

   9. Taylor expanded in b around 0 88.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.15 \cdot 10^{+152}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{\frac{y \cdot e^{b}}{{a}^{\left(t + -1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{a}}{y}\\ \end{array} \]

Alternative 6: 80.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+166}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.15e+166)
   (/ (* x (/ (pow z y) a)) y)
   (if (<= y 3.7e-12)
     (/ (* x (/ (pow a (+ t -1.0)) (exp b))) y)
     (/ (/ (* x (pow z y)) a) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.15e+166) {
		tmp = (x * (pow(z, y) / a)) / y;
	} else if (y <= 3.7e-12) {
		tmp = (x * (pow(a, (t + -1.0)) / exp(b))) / y;
	} else {
		tmp = ((x * pow(z, y)) / a) / y;
	}
	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 (y <= (-1.15d+166)) then
        tmp = (x * ((z ** y) / a)) / y
    else if (y <= 3.7d-12) then
        tmp = (x * ((a ** (t + (-1.0d0))) / exp(b))) / y
    else
        tmp = ((x * (z ** y)) / a) / y
    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 (y <= -1.15e+166) {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	} else if (y <= 3.7e-12) {
		tmp = (x * (Math.pow(a, (t + -1.0)) / Math.exp(b))) / y;
	} else {
		tmp = ((x * Math.pow(z, y)) / a) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.15e+166:
		tmp = (x * (math.pow(z, y) / a)) / y
	elif y <= 3.7e-12:
		tmp = (x * (math.pow(a, (t + -1.0)) / math.exp(b))) / y
	else:
		tmp = ((x * math.pow(z, y)) / a) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.15e+166)
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	elseif (y <= 3.7e-12)
		tmp = Float64(Float64(x * Float64((a ^ Float64(t + -1.0)) / exp(b))) / y);
	else
		tmp = Float64(Float64(Float64(x * (z ^ y)) / a) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.15e+166)
		tmp = (x * ((z ^ y) / a)) / y;
	elseif (y <= 3.7e-12)
		tmp = (x * ((a ^ (t + -1.0)) / exp(b))) / y;
	else
		tmp = ((x * (z ^ y)) / a) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.15e+166], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 3.7e-12], N[(N[(x * N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] / N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15000000000000004e166

   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. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. fma-def 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 94.7%

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

   5. Taylor expanded in b around 0 94.7%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}}{y} \]
   6. Step-by-step derivation

      1. +-commutative 94.7%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 94.7%

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

      3. sub-neg 94.7%

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

      4. exp-diff 94.7%

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

      5. *-commutative 94.7%

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

      6. exp-to-pow 94.7%

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

      7. rem-exp-log 94.7%

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

      8. *-commutative 94.7%

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

   7. Simplified 94.7%

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

   if -1.15000000000000004e166 < y < 3.69999999999999999e-12

   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. Taylor expanded in y around 0 95.3%

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

      1. div-exp 86.6%

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

      2. exp-to-pow 87.2%

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

      3. sub-neg 87.2%

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

      4. metadata-eval 87.2%

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

   4. Simplified 87.2%

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

   if 3.69999999999999999e-12 < y

   1. Initial program 99.6%

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

      1. expm1-log1p-u 84.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right)\right)}}{y} \]

      2. expm1-udef 81.6%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right)} - 1}}{y} \]

   3. Applied egg-rr 46.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(\frac{{a}^{t}}{a} \cdot \frac{{z}^{y}}{e^{b}}\right)\right)} - 1}}{y} \]
   4. Step-by-step derivation

      1. expm1-def 49.1%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\frac{{a}^{t}}{a} \cdot \frac{{z}^{y}}{e^{b}}\right)\right)\right)}}{y} \]

      2. expm1-log1p 61.0%

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

      3. *-commutative 61.0%

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

      4. associate-*l/ 61.0%

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

      5. associate-*r/ 61.0%

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

      6. associate-/l/ 61.0%

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

      7. *-commutative 61.0%

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

   5. Simplified 61.0%

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

   6. Taylor expanded in t around 0 74.6%

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

      1. associate-/r* 74.6%

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

   8. Simplified 74.6%

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

   9. Taylor expanded in b around 0 88.4%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+166}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{a}}{y}\\ \end{array} \]

Alternative 7: 73.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ t_2 := \frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ t_3 := \frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{if}\;b \leq -340:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -2.05 \cdot 10^{-202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-179}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{+29}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ (pow z y) a) (/ x y)))
        (t_2 (/ x (* a (* y (exp b)))))
        (t_3 (/ (* x (pow a (+ t -1.0))) y)))
   (if (<= b -340.0)
     t_2
     (if (<= b -2.05e-202)
       t_1
       (if (<= b 1.1e-179)
         t_3
         (if (<= b 9e-135) t_1 (if (<= b 8.8e+29) t_3 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (pow(z, y) / a) * (x / y);
	double t_2 = x / (a * (y * exp(b)));
	double t_3 = (x * pow(a, (t + -1.0))) / y;
	double tmp;
	if (b <= -340.0) {
		tmp = t_2;
	} else if (b <= -2.05e-202) {
		tmp = t_1;
	} else if (b <= 1.1e-179) {
		tmp = t_3;
	} else if (b <= 9e-135) {
		tmp = t_1;
	} else if (b <= 8.8e+29) {
		tmp = t_3;
	} 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) :: t_3
    real(8) :: tmp
    t_1 = ((z ** y) / a) * (x / y)
    t_2 = x / (a * (y * exp(b)))
    t_3 = (x * (a ** (t + (-1.0d0)))) / y
    if (b <= (-340.0d0)) then
        tmp = t_2
    else if (b <= (-2.05d-202)) then
        tmp = t_1
    else if (b <= 1.1d-179) then
        tmp = t_3
    else if (b <= 9d-135) then
        tmp = t_1
    else if (b <= 8.8d+29) then
        tmp = t_3
    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(z, y) / a) * (x / y);
	double t_2 = x / (a * (y * Math.exp(b)));
	double t_3 = (x * Math.pow(a, (t + -1.0))) / y;
	double tmp;
	if (b <= -340.0) {
		tmp = t_2;
	} else if (b <= -2.05e-202) {
		tmp = t_1;
	} else if (b <= 1.1e-179) {
		tmp = t_3;
	} else if (b <= 9e-135) {
		tmp = t_1;
	} else if (b <= 8.8e+29) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.pow(z, y) / a) * (x / y)
	t_2 = x / (a * (y * math.exp(b)))
	t_3 = (x * math.pow(a, (t + -1.0))) / y
	tmp = 0
	if b <= -340.0:
		tmp = t_2
	elif b <= -2.05e-202:
		tmp = t_1
	elif b <= 1.1e-179:
		tmp = t_3
	elif b <= 9e-135:
		tmp = t_1
	elif b <= 8.8e+29:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64((z ^ y) / a) * Float64(x / y))
	t_2 = Float64(x / Float64(a * Float64(y * exp(b))))
	t_3 = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y)
	tmp = 0.0
	if (b <= -340.0)
		tmp = t_2;
	elseif (b <= -2.05e-202)
		tmp = t_1;
	elseif (b <= 1.1e-179)
		tmp = t_3;
	elseif (b <= 9e-135)
		tmp = t_1;
	elseif (b <= 8.8e+29)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((z ^ y) / a) * (x / y);
	t_2 = x / (a * (y * exp(b)));
	t_3 = (x * (a ^ (t + -1.0))) / y;
	tmp = 0.0;
	if (b <= -340.0)
		tmp = t_2;
	elseif (b <= -2.05e-202)
		tmp = t_1;
	elseif (b <= 1.1e-179)
		tmp = t_3;
	elseif (b <= 9e-135)
		tmp = t_1;
	elseif (b <= 8.8e+29)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[b, -340.0], t$95$2, If[LessEqual[b, -2.05e-202], t$95$1, If[LessEqual[b, 1.1e-179], t$95$3, If[LessEqual[b, 9e-135], t$95$1, If[LessEqual[b, 8.8e+29], t$95$3, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -340 or 8.8000000000000005e29 < 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. Step-by-step derivation

      1. associate-*l/ 91.1%

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

      2. *-commutative 91.1%

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

      3. +-commutative 91.1%

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

      4. associate--l+ 91.1%

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

      5. exp-sum 74.0%

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

      6. *-commutative 74.0%

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

      7. exp-to-pow 74.0%

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

      8. sub-neg 74.0%

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

      9. metadata-eval 74.0%

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

      10. exp-diff 58.5%

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

      11. *-commutative 58.5%

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

      12. exp-to-pow 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in t around 0 73.3%

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

      1. *-commutative 73.3%

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

      2. times-frac 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in y around 0 84.8%

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

   if -340 < b < -2.0500000000000002e-202 or 1.10000000000000002e-179 < b < 8.99999999999999975e-135

   1. Initial program 98.3%

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

      1. associate-/l* 95.3%

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

      2. fma-def 95.3%

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

      3. sub-neg 95.3%

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

      4. metadata-eval 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in t around 0 86.0%

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

   5. Taylor expanded in b around 0 86.0%

      \[\leadsto \color{blue}{\frac{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}{y}} \]
   6. Step-by-step derivation

      1. +-commutative 86.0%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 86.0%

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

      3. sub-neg 86.0%

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

      4. exp-diff 85.9%

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

      5. *-commutative 85.9%

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

      6. exp-to-pow 85.9%

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

      7. rem-exp-log 87.3%

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

      8. *-commutative 87.3%

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

      9. associate-*l/ 87.4%

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

      10. *-commutative 87.4%

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

      11. associate-/r* 67.3%

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

      12. *-commutative 67.3%

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

      13. times-frac 76.9%

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

   7. Simplified 76.9%

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

   if -2.0500000000000002e-202 < b < 1.10000000000000002e-179 or 8.99999999999999975e-135 < b < 8.8000000000000005e29

   1. Initial program 97.6%

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

   2. Taylor expanded in y around 0 77.8%

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

      1. div-exp 75.5%

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

      2. exp-to-pow 75.9%

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

      3. sub-neg 75.9%

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

      4. metadata-eval 75.9%

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

   4. Simplified 75.9%

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

   5. Taylor expanded in b around 0 77.8%

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

      1. exp-to-pow 78.3%

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

      2. sub-neg 78.3%

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

      3. metadata-eval 78.3%

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

      4. +-commutative 78.3%

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

   7. Simplified 78.3%

      \[\leadsto \frac{\color{blue}{x \cdot {a}^{\left(-1 + t\right)}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -340:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;b \leq -2.05 \cdot 10^{-202}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-179}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-135}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 8: 73.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ t_2 := \frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ t_3 := \frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{if}\;y \leq -6 \cdot 10^{+147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-22}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-12}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* a (* y (exp b)))))
        (t_2 (/ (* x (/ (pow z y) a)) y))
        (t_3 (/ (* x (pow a (+ t -1.0))) y)))
   (if (<= y -6e+147)
     t_2
     (if (<= y -3.1e+21)
       t_1
       (if (<= y -9e-22)
         t_3
         (if (<= y 2.2e-277) t_1 (if (<= y 2.4e-12) t_3 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a * (y * exp(b)));
	double t_2 = (x * (pow(z, y) / a)) / y;
	double t_3 = (x * pow(a, (t + -1.0))) / y;
	double tmp;
	if (y <= -6e+147) {
		tmp = t_2;
	} else if (y <= -3.1e+21) {
		tmp = t_1;
	} else if (y <= -9e-22) {
		tmp = t_3;
	} else if (y <= 2.2e-277) {
		tmp = t_1;
	} else if (y <= 2.4e-12) {
		tmp = t_3;
	} 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) :: t_3
    real(8) :: tmp
    t_1 = x / (a * (y * exp(b)))
    t_2 = (x * ((z ** y) / a)) / y
    t_3 = (x * (a ** (t + (-1.0d0)))) / y
    if (y <= (-6d+147)) then
        tmp = t_2
    else if (y <= (-3.1d+21)) then
        tmp = t_1
    else if (y <= (-9d-22)) then
        tmp = t_3
    else if (y <= 2.2d-277) then
        tmp = t_1
    else if (y <= 2.4d-12) then
        tmp = t_3
    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 / (a * (y * Math.exp(b)));
	double t_2 = (x * (Math.pow(z, y) / a)) / y;
	double t_3 = (x * Math.pow(a, (t + -1.0))) / y;
	double tmp;
	if (y <= -6e+147) {
		tmp = t_2;
	} else if (y <= -3.1e+21) {
		tmp = t_1;
	} else if (y <= -9e-22) {
		tmp = t_3;
	} else if (y <= 2.2e-277) {
		tmp = t_1;
	} else if (y <= 2.4e-12) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (a * (y * math.exp(b)))
	t_2 = (x * (math.pow(z, y) / a)) / y
	t_3 = (x * math.pow(a, (t + -1.0))) / y
	tmp = 0
	if y <= -6e+147:
		tmp = t_2
	elif y <= -3.1e+21:
		tmp = t_1
	elif y <= -9e-22:
		tmp = t_3
	elif y <= 2.2e-277:
		tmp = t_1
	elif y <= 2.4e-12:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a * Float64(y * exp(b))))
	t_2 = Float64(Float64(x * Float64((z ^ y) / a)) / y)
	t_3 = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y)
	tmp = 0.0
	if (y <= -6e+147)
		tmp = t_2;
	elseif (y <= -3.1e+21)
		tmp = t_1;
	elseif (y <= -9e-22)
		tmp = t_3;
	elseif (y <= 2.2e-277)
		tmp = t_1;
	elseif (y <= 2.4e-12)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (a * (y * exp(b)));
	t_2 = (x * ((z ^ y) / a)) / y;
	t_3 = (x * (a ^ (t + -1.0))) / y;
	tmp = 0.0;
	if (y <= -6e+147)
		tmp = t_2;
	elseif (y <= -3.1e+21)
		tmp = t_1;
	elseif (y <= -9e-22)
		tmp = t_3;
	elseif (y <= 2.2e-277)
		tmp = t_1;
	elseif (y <= 2.4e-12)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -6e+147], t$95$2, If[LessEqual[y, -3.1e+21], t$95$1, If[LessEqual[y, -9e-22], t$95$3, If[LessEqual[y, 2.2e-277], t$95$1, If[LessEqual[y, 2.4e-12], t$95$3, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.99999999999999987e147 or 2.39999999999999987e-12 < y

   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. Step-by-step derivation

      1. associate-/l* 99.8%

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

      2. fma-def 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 95.0%

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

   5. Taylor expanded in b around 0 90.2%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}}{y} \]
   6. Step-by-step derivation

      1. +-commutative 90.2%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 90.2%

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

      3. sub-neg 90.2%

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

      4. exp-diff 90.2%

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

      5. *-commutative 90.2%

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

      6. exp-to-pow 90.2%

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

      7. rem-exp-log 90.4%

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

      8. *-commutative 90.4%

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

   7. Simplified 90.4%

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

   if -5.99999999999999987e147 < y < -3.1e21 or -8.99999999999999973e-22 < y < 2.19999999999999996e-277

   1. Initial program 97.7%

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

      1. associate-*l/ 95.7%

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

      2. *-commutative 95.7%

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

      3. +-commutative 95.7%

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

      4. associate--l+ 95.7%

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

      5. exp-sum 85.5%

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

      6. *-commutative 85.5%

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

      7. exp-to-pow 86.2%

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

      8. sub-neg 86.2%

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

      9. metadata-eval 86.2%

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

      10. exp-diff 79.4%

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

      11. *-commutative 79.4%

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

      12. exp-to-pow 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in t around 0 74.4%

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

      1. *-commutative 74.4%

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

      2. times-frac 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in y around 0 79.3%

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

   if -3.1e21 < y < -8.99999999999999973e-22 or 2.19999999999999996e-277 < y < 2.39999999999999987e-12

   1. Initial program 99.1%

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

   2. Taylor expanded in y around 0 99.1%

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

      1. div-exp 85.5%

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

      2. exp-to-pow 86.1%

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

      3. sub-neg 86.1%

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

      4. metadata-eval 86.1%

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

   4. Simplified 86.1%

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

   5. Taylor expanded in b around 0 79.8%

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

      1. exp-to-pow 80.5%

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

      2. sub-neg 80.5%

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

      3. metadata-eval 80.5%

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

      4. +-commutative 80.5%

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

   7. Simplified 80.5%

      \[\leadsto \frac{\color{blue}{x \cdot {a}^{\left(-1 + t\right)}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+147}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-22}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-277}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \end{array} \]

Alternative 9: 73.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ t_2 := \frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{if}\;y \leq -6 \cdot 10^{+147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-276}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{x \cdot {a}^{t}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* a (* y (exp b))))) (t_2 (/ (* x (/ (pow z y) a)) y)))
   (if (<= y -6e+147)
     t_2
     (if (<= y -1.75e+18)
       t_1
       (if (<= y -3.7e-22)
         (/ (* x (pow a (+ t -1.0))) y)
         (if (<= y 2e-276)
           t_1
           (if (<= y 3.7e-12) (/ (/ (* x (pow a t)) a) y) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a * (y * exp(b)));
	double t_2 = (x * (pow(z, y) / a)) / y;
	double tmp;
	if (y <= -6e+147) {
		tmp = t_2;
	} else if (y <= -1.75e+18) {
		tmp = t_1;
	} else if (y <= -3.7e-22) {
		tmp = (x * pow(a, (t + -1.0))) / y;
	} else if (y <= 2e-276) {
		tmp = t_1;
	} else if (y <= 3.7e-12) {
		tmp = ((x * pow(a, t)) / a) / 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 = x / (a * (y * exp(b)))
    t_2 = (x * ((z ** y) / a)) / y
    if (y <= (-6d+147)) then
        tmp = t_2
    else if (y <= (-1.75d+18)) then
        tmp = t_1
    else if (y <= (-3.7d-22)) then
        tmp = (x * (a ** (t + (-1.0d0)))) / y
    else if (y <= 2d-276) then
        tmp = t_1
    else if (y <= 3.7d-12) then
        tmp = ((x * (a ** t)) / a) / 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 = x / (a * (y * Math.exp(b)));
	double t_2 = (x * (Math.pow(z, y) / a)) / y;
	double tmp;
	if (y <= -6e+147) {
		tmp = t_2;
	} else if (y <= -1.75e+18) {
		tmp = t_1;
	} else if (y <= -3.7e-22) {
		tmp = (x * Math.pow(a, (t + -1.0))) / y;
	} else if (y <= 2e-276) {
		tmp = t_1;
	} else if (y <= 3.7e-12) {
		tmp = ((x * Math.pow(a, t)) / a) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (a * (y * math.exp(b)))
	t_2 = (x * (math.pow(z, y) / a)) / y
	tmp = 0
	if y <= -6e+147:
		tmp = t_2
	elif y <= -1.75e+18:
		tmp = t_1
	elif y <= -3.7e-22:
		tmp = (x * math.pow(a, (t + -1.0))) / y
	elif y <= 2e-276:
		tmp = t_1
	elif y <= 3.7e-12:
		tmp = ((x * math.pow(a, t)) / a) / y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a * Float64(y * exp(b))))
	t_2 = Float64(Float64(x * Float64((z ^ y) / a)) / y)
	tmp = 0.0
	if (y <= -6e+147)
		tmp = t_2;
	elseif (y <= -1.75e+18)
		tmp = t_1;
	elseif (y <= -3.7e-22)
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	elseif (y <= 2e-276)
		tmp = t_1;
	elseif (y <= 3.7e-12)
		tmp = Float64(Float64(Float64(x * (a ^ t)) / a) / y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (a * (y * exp(b)));
	t_2 = (x * ((z ^ y) / a)) / y;
	tmp = 0.0;
	if (y <= -6e+147)
		tmp = t_2;
	elseif (y <= -1.75e+18)
		tmp = t_1;
	elseif (y <= -3.7e-22)
		tmp = (x * (a ^ (t + -1.0))) / y;
	elseif (y <= 2e-276)
		tmp = t_1;
	elseif (y <= 3.7e-12)
		tmp = ((x * (a ^ t)) / a) / y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -6e+147], t$95$2, If[LessEqual[y, -1.75e+18], t$95$1, If[LessEqual[y, -3.7e-22], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 2e-276], t$95$1, If[LessEqual[y, 3.7e-12], N[(N[(N[(x * N[Power[a, t], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.99999999999999987e147 or 3.69999999999999999e-12 < y

   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. Step-by-step derivation

      1. associate-/l* 99.8%

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

      2. fma-def 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 95.0%

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

   5. Taylor expanded in b around 0 90.2%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}}{y} \]
   6. Step-by-step derivation

      1. +-commutative 90.2%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 90.2%

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

      3. sub-neg 90.2%

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

      4. exp-diff 90.2%

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

      5. *-commutative 90.2%

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

      6. exp-to-pow 90.2%

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

      7. rem-exp-log 90.4%

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

      8. *-commutative 90.4%

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

   7. Simplified 90.4%

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

   if -5.99999999999999987e147 < y < -1.75e18 or -3.7e-22 < y < 2e-276

   1. Initial program 97.7%

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

      1. associate-*l/ 95.7%

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

      2. *-commutative 95.7%

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

      3. +-commutative 95.7%

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

      4. associate--l+ 95.7%

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

      5. exp-sum 85.5%

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

      6. *-commutative 85.5%

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

      7. exp-to-pow 86.2%

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

      8. sub-neg 86.2%

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

      9. metadata-eval 86.2%

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

      10. exp-diff 79.4%

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

      11. *-commutative 79.4%

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

      12. exp-to-pow 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in t around 0 74.4%

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

      1. *-commutative 74.4%

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

      2. times-frac 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in y around 0 79.3%

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

   if -1.75e18 < y < -3.7e-22

   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. Taylor expanded in y around 0 100.0%

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

      1. div-exp 87.5%

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

      2. exp-to-pow 87.5%

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

      3. sub-neg 87.5%

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

      4. metadata-eval 87.5%

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

   4. Simplified 87.5%

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

   5. Taylor expanded in b around 0 100.0%

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

      1. exp-to-pow 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   if 2e-276 < y < 3.69999999999999999e-12

   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. Step-by-step derivation

      1. expm1-log1p-u 73.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right)\right)}}{y} \]

      2. expm1-udef 67.3%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right)} - 1}}{y} \]

   3. Applied egg-rr 58.7%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(\frac{{a}^{t}}{a} \cdot \frac{{z}^{y}}{e^{b}}\right)\right)} - 1}}{y} \]
   4. Step-by-step derivation

      1. expm1-def 65.3%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\frac{{a}^{t}}{a} \cdot \frac{{z}^{y}}{e^{b}}\right)\right)\right)}}{y} \]

      2. expm1-log1p 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-*l/ 86.2%

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

      5. associate-*r/ 86.2%

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

      6. associate-/l/ 86.2%

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

      7. *-commutative 86.2%

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

   5. Simplified 86.2%

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

   6. Taylor expanded in y around 0 86.2%

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

      1. associate-/r* 79.3%

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

   8. Simplified 79.3%

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

   9. Taylor expanded in b around 0 78.0%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot {a}^{t}}{a}}}{y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+147}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-276}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{x \cdot {a}^{t}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \end{array} \]

Alternative 10: 73.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{if}\;y \leq -6 \cdot 10^{+147}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-281}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{x \cdot {a}^{t}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* a (* y (exp b))))))
   (if (<= y -6e+147)
     (/ (* x (/ (pow z y) a)) y)
     (if (<= y -3.2e+17)
       t_1
       (if (<= y -3.7e-22)
         (/ (* x (pow a (+ t -1.0))) y)
         (if (<= y 7.5e-281)
           t_1
           (if (<= y 3.7e-12)
             (/ (/ (* x (pow a t)) a) y)
             (/ (/ (* x (pow z y)) a) y))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a * (y * exp(b)));
	double tmp;
	if (y <= -6e+147) {
		tmp = (x * (pow(z, y) / a)) / y;
	} else if (y <= -3.2e+17) {
		tmp = t_1;
	} else if (y <= -3.7e-22) {
		tmp = (x * pow(a, (t + -1.0))) / y;
	} else if (y <= 7.5e-281) {
		tmp = t_1;
	} else if (y <= 3.7e-12) {
		tmp = ((x * pow(a, t)) / a) / y;
	} else {
		tmp = ((x * pow(z, y)) / a) / y;
	}
	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 * (y * exp(b)))
    if (y <= (-6d+147)) then
        tmp = (x * ((z ** y) / a)) / y
    else if (y <= (-3.2d+17)) then
        tmp = t_1
    else if (y <= (-3.7d-22)) then
        tmp = (x * (a ** (t + (-1.0d0)))) / y
    else if (y <= 7.5d-281) then
        tmp = t_1
    else if (y <= 3.7d-12) then
        tmp = ((x * (a ** t)) / a) / y
    else
        tmp = ((x * (z ** y)) / a) / y
    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 / (a * (y * Math.exp(b)));
	double tmp;
	if (y <= -6e+147) {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	} else if (y <= -3.2e+17) {
		tmp = t_1;
	} else if (y <= -3.7e-22) {
		tmp = (x * Math.pow(a, (t + -1.0))) / y;
	} else if (y <= 7.5e-281) {
		tmp = t_1;
	} else if (y <= 3.7e-12) {
		tmp = ((x * Math.pow(a, t)) / a) / y;
	} else {
		tmp = ((x * Math.pow(z, y)) / a) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (a * (y * math.exp(b)))
	tmp = 0
	if y <= -6e+147:
		tmp = (x * (math.pow(z, y) / a)) / y
	elif y <= -3.2e+17:
		tmp = t_1
	elif y <= -3.7e-22:
		tmp = (x * math.pow(a, (t + -1.0))) / y
	elif y <= 7.5e-281:
		tmp = t_1
	elif y <= 3.7e-12:
		tmp = ((x * math.pow(a, t)) / a) / y
	else:
		tmp = ((x * math.pow(z, y)) / a) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a * Float64(y * exp(b))))
	tmp = 0.0
	if (y <= -6e+147)
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	elseif (y <= -3.2e+17)
		tmp = t_1;
	elseif (y <= -3.7e-22)
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	elseif (y <= 7.5e-281)
		tmp = t_1;
	elseif (y <= 3.7e-12)
		tmp = Float64(Float64(Float64(x * (a ^ t)) / a) / y);
	else
		tmp = Float64(Float64(Float64(x * (z ^ y)) / a) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (a * (y * exp(b)));
	tmp = 0.0;
	if (y <= -6e+147)
		tmp = (x * ((z ^ y) / a)) / y;
	elseif (y <= -3.2e+17)
		tmp = t_1;
	elseif (y <= -3.7e-22)
		tmp = (x * (a ^ (t + -1.0))) / y;
	elseif (y <= 7.5e-281)
		tmp = t_1;
	elseif (y <= 3.7e-12)
		tmp = ((x * (a ^ t)) / a) / y;
	else
		tmp = ((x * (z ^ y)) / a) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6e+147], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, -3.2e+17], t$95$1, If[LessEqual[y, -3.7e-22], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 7.5e-281], t$95$1, If[LessEqual[y, 3.7e-12], N[(N[(N[(x * N[Power[a, t], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -5.99999999999999987e147

   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. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. fma-def 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 93.1%

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

   5. Taylor expanded in b around 0 93.1%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}}{y} \]
   6. Step-by-step derivation

      1. +-commutative 93.1%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 93.1%

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

      3. sub-neg 93.1%

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

      4. exp-diff 93.1%

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

      5. *-commutative 93.1%

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

      6. exp-to-pow 93.1%

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

      7. rem-exp-log 93.1%

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

      8. *-commutative 93.1%

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

   7. Simplified 93.1%

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

   if -5.99999999999999987e147 < y < -3.2e17 or -3.7e-22 < y < 7.49999999999999968e-281

   1. Initial program 97.7%

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

      1. associate-*l/ 95.7%

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

      2. *-commutative 95.7%

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

      3. +-commutative 95.7%

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

      4. associate--l+ 95.7%

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

      5. exp-sum 85.5%

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

      6. *-commutative 85.5%

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

      7. exp-to-pow 86.2%

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

      8. sub-neg 86.2%

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

      9. metadata-eval 86.2%

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

      10. exp-diff 79.4%

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

      11. *-commutative 79.4%

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

      12. exp-to-pow 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in t around 0 74.4%

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

      1. *-commutative 74.4%

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

      2. times-frac 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in y around 0 79.3%

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

   if -3.2e17 < y < -3.7e-22

   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. Taylor expanded in y around 0 100.0%

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

      1. div-exp 87.5%

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

      2. exp-to-pow 87.5%

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

      3. sub-neg 87.5%

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

      4. metadata-eval 87.5%

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

   4. Simplified 87.5%

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

   5. Taylor expanded in b around 0 100.0%

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

      1. exp-to-pow 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   if 7.49999999999999968e-281 < y < 3.69999999999999999e-12

   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. Step-by-step derivation

      1. expm1-log1p-u 73.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right)\right)}}{y} \]

      2. expm1-udef 67.3%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right)} - 1}}{y} \]

   3. Applied egg-rr 58.7%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(\frac{{a}^{t}}{a} \cdot \frac{{z}^{y}}{e^{b}}\right)\right)} - 1}}{y} \]
   4. Step-by-step derivation

      1. expm1-def 65.3%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\frac{{a}^{t}}{a} \cdot \frac{{z}^{y}}{e^{b}}\right)\right)\right)}}{y} \]

      2. expm1-log1p 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-*l/ 86.2%

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

      5. associate-*r/ 86.2%

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

      6. associate-/l/ 86.2%

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

      7. *-commutative 86.2%

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

   5. Simplified 86.2%

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

   6. Taylor expanded in y around 0 86.2%

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

      1. associate-/r* 79.3%

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

   8. Simplified 79.3%

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

   9. Taylor expanded in b around 0 78.0%

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

   if 3.69999999999999999e-12 < y

   1. Initial program 99.6%

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

      1. expm1-log1p-u 84.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right)\right)}}{y} \]

      2. expm1-udef 81.6%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right)} - 1}}{y} \]

   3. Applied egg-rr 46.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(\frac{{a}^{t}}{a} \cdot \frac{{z}^{y}}{e^{b}}\right)\right)} - 1}}{y} \]
   4. Step-by-step derivation

      1. expm1-def 49.1%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\frac{{a}^{t}}{a} \cdot \frac{{z}^{y}}{e^{b}}\right)\right)\right)}}{y} \]

      2. expm1-log1p 61.0%

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

      3. *-commutative 61.0%

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

      4. associate-*l/ 61.0%

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

      5. associate-*r/ 61.0%

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

      6. associate-/l/ 61.0%

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

      7. *-commutative 61.0%

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

   5. Simplified 61.0%

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

   6. Taylor expanded in t around 0 74.6%

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

      1. associate-/r* 74.6%

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

   8. Simplified 74.6%

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

   9. Taylor expanded in b around 0 88.4%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot {z}^{y}}{a}}}{y} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+147}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-281}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{x \cdot {a}^{t}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{a}}{y}\\ \end{array} \]

Alternative 11: 72.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -850 \lor \neg \left(b \leq 62000000000000\right):\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -850.0) (not (<= b 62000000000000.0)))
   (/ x (* a (* y (exp b))))
   (* (/ (pow z y) a) (/ x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -850.0) || !(b <= 62000000000000.0)) {
		tmp = x / (a * (y * exp(b)));
	} else {
		tmp = (pow(z, y) / a) * (x / y);
	}
	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 <= (-850.0d0)) .or. (.not. (b <= 62000000000000.0d0))) then
        tmp = x / (a * (y * exp(b)))
    else
        tmp = ((z ** y) / a) * (x / y)
    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 <= -850.0) || !(b <= 62000000000000.0)) {
		tmp = x / (a * (y * Math.exp(b)));
	} else {
		tmp = (Math.pow(z, y) / a) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -850.0) or not (b <= 62000000000000.0):
		tmp = x / (a * (y * math.exp(b)))
	else:
		tmp = (math.pow(z, y) / a) * (x / y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -850.0) || !(b <= 62000000000000.0))
		tmp = Float64(x / Float64(a * Float64(y * exp(b))));
	else
		tmp = Float64(Float64((z ^ y) / a) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -850.0) || ~((b <= 62000000000000.0)))
		tmp = x / (a * (y * exp(b)));
	else
		tmp = ((z ^ y) / a) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -850.0], N[Not[LessEqual[b, 62000000000000.0]], $MachinePrecision]], N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -850 or 6.2e13 < 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. Step-by-step derivation

      1. associate-*l/ 91.3%

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

      2. *-commutative 91.3%

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

      3. +-commutative 91.3%

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

      4. associate--l+ 91.3%

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

      5. exp-sum 74.6%

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

      6. *-commutative 74.6%

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

      7. exp-to-pow 74.6%

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

      8. sub-neg 74.6%

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

      9. metadata-eval 74.6%

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

      10. exp-diff 58.7%

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

      11. *-commutative 58.7%

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

      12. exp-to-pow 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in t around 0 73.1%

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

      1. *-commutative 73.1%

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

      2. times-frac 73.1%

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

   6. Simplified 73.1%

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

   7. Taylor expanded in y around 0 84.4%

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

   if -850 < b < 6.2e13

   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. Step-by-step derivation

      1. associate-/l* 97.0%

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

      2. fma-def 97.0%

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

      3. sub-neg 97.0%

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

      4. metadata-eval 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in t around 0 72.2%

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

   5. Taylor expanded in b around 0 73.0%

      \[\leadsto \color{blue}{\frac{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}{y}} \]
   6. Step-by-step derivation

      1. +-commutative 73.0%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 73.0%

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

      3. sub-neg 73.0%

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

      4. exp-diff 73.0%

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

      5. *-commutative 73.0%

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

      6. exp-to-pow 73.0%

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

      7. rem-exp-log 73.8%

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

      8. *-commutative 73.8%

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

      9. associate-*l/ 73.9%

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

      10. *-commutative 73.9%

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

      11. associate-/r* 64.1%

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

      12. *-commutative 64.1%

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

      13. times-frac 69.2%

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

   7. Simplified 69.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -850 \lor \neg \left(b \leq 62000000000000\right):\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 12: 59.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{x}{a \cdot \left(y \cdot e^{b}\right)} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ x (* a (* y (exp b)))))

C

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

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 / (a * (y * exp(b)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l/ 92.0%

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

   2. *-commutative 92.0%

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

   3. +-commutative 92.0%

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

   4. associate--l+ 92.0%

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

   5. exp-sum 77.1%

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

   6. *-commutative 77.1%

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

   7. exp-to-pow 77.6%

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

   8. sub-neg 77.6%

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

   9. metadata-eval 77.6%

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

   10. exp-diff 69.8%

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

   11. *-commutative 69.8%

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

   12. exp-to-pow 69.8%

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

3. Simplified 69.8%

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

4. Taylor expanded in t around 0 68.1%

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

   1. *-commutative 68.1%

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

   2. times-frac 70.7%

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

6. Simplified 70.7%

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

7. Taylor expanded in y around 0 59.8%

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

8. Final simplification 59.8%

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

Alternative 13: 35.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + y \cdot b\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+200}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{x}{t_1}}}\\ \mathbf{elif}\;y \leq -120000000000:\\ \;\;\;\;\left(b \cdot b\right) \cdot \frac{x \cdot 0.5}{y \cdot a} - \frac{x}{y \cdot \frac{a}{b}}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-244}:\\ \;\;\;\;\frac{x - x \cdot b}{y \cdot a} + \frac{x}{y} \cdot \frac{b \cdot b}{a}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-255}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* y b))))
   (if (<= y -5.5e+200)
     (/ 1.0 (/ a (/ x t_1)))
     (if (<= y -120000000000.0)
       (- (* (* b b) (/ (* x 0.5) (* y a))) (/ x (* y (/ a b))))
       (if (<= y -1.7e-244)
         (+ (/ (- x (* x b)) (* y a)) (* (/ x y) (/ (* b b) a)))
         (if (<= y 2.15e-255)
           (/ x (* y (+ a (* a b))))
           (if (<= y 2.55e-126)
             (/ (/ (* x (- 1.0 b)) y) a)
             (/ x (* a t_1)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (y * b);
	double tmp;
	if (y <= -5.5e+200) {
		tmp = 1.0 / (a / (x / t_1));
	} else if (y <= -120000000000.0) {
		tmp = ((b * b) * ((x * 0.5) / (y * a))) - (x / (y * (a / b)));
	} else if (y <= -1.7e-244) {
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a));
	} else if (y <= 2.15e-255) {
		tmp = x / (y * (a + (a * b)));
	} else if (y <= 2.55e-126) {
		tmp = ((x * (1.0 - b)) / y) / a;
	} else {
		tmp = x / (a * 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 = y + (y * b)
    if (y <= (-5.5d+200)) then
        tmp = 1.0d0 / (a / (x / t_1))
    else if (y <= (-120000000000.0d0)) then
        tmp = ((b * b) * ((x * 0.5d0) / (y * a))) - (x / (y * (a / b)))
    else if (y <= (-1.7d-244)) then
        tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a))
    else if (y <= 2.15d-255) then
        tmp = x / (y * (a + (a * b)))
    else if (y <= 2.55d-126) then
        tmp = ((x * (1.0d0 - b)) / y) / a
    else
        tmp = x / (a * 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 = y + (y * b);
	double tmp;
	if (y <= -5.5e+200) {
		tmp = 1.0 / (a / (x / t_1));
	} else if (y <= -120000000000.0) {
		tmp = ((b * b) * ((x * 0.5) / (y * a))) - (x / (y * (a / b)));
	} else if (y <= -1.7e-244) {
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a));
	} else if (y <= 2.15e-255) {
		tmp = x / (y * (a + (a * b)));
	} else if (y <= 2.55e-126) {
		tmp = ((x * (1.0 - b)) / y) / a;
	} else {
		tmp = x / (a * t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y + (y * b)
	tmp = 0
	if y <= -5.5e+200:
		tmp = 1.0 / (a / (x / t_1))
	elif y <= -120000000000.0:
		tmp = ((b * b) * ((x * 0.5) / (y * a))) - (x / (y * (a / b)))
	elif y <= -1.7e-244:
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a))
	elif y <= 2.15e-255:
		tmp = x / (y * (a + (a * b)))
	elif y <= 2.55e-126:
		tmp = ((x * (1.0 - b)) / y) / a
	else:
		tmp = x / (a * t_1)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(y * b))
	tmp = 0.0
	if (y <= -5.5e+200)
		tmp = Float64(1.0 / Float64(a / Float64(x / t_1)));
	elseif (y <= -120000000000.0)
		tmp = Float64(Float64(Float64(b * b) * Float64(Float64(x * 0.5) / Float64(y * a))) - Float64(x / Float64(y * Float64(a / b))));
	elseif (y <= -1.7e-244)
		tmp = Float64(Float64(Float64(x - Float64(x * b)) / Float64(y * a)) + Float64(Float64(x / y) * Float64(Float64(b * b) / a)));
	elseif (y <= 2.15e-255)
		tmp = Float64(x / Float64(y * Float64(a + Float64(a * b))));
	elseif (y <= 2.55e-126)
		tmp = Float64(Float64(Float64(x * Float64(1.0 - b)) / y) / a);
	else
		tmp = Float64(x / Float64(a * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y + (y * b);
	tmp = 0.0;
	if (y <= -5.5e+200)
		tmp = 1.0 / (a / (x / t_1));
	elseif (y <= -120000000000.0)
		tmp = ((b * b) * ((x * 0.5) / (y * a))) - (x / (y * (a / b)));
	elseif (y <= -1.7e-244)
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a));
	elseif (y <= 2.15e-255)
		tmp = x / (y * (a + (a * b)));
	elseif (y <= 2.55e-126)
		tmp = ((x * (1.0 - b)) / y) / a;
	else
		tmp = x / (a * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(y * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e+200], N[(1.0 / N[(a / N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -120000000000.0], N[(N[(N[(b * b), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(y * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.7e-244], N[(N[(N[(x - N[(x * b), $MachinePrecision]), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.15e-255], N[(x / N[(y * N[(a + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.55e-126], N[(N[(N[(x * N[(1.0 - b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / a), $MachinePrecision], N[(x / N[(a * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y < -5.5e200

   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. Step-by-step derivation

      1. associate-*l/ 93.1%

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

      2. *-commutative 93.1%

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

      3. +-commutative 93.1%

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

      4. associate--l+ 93.1%

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

      5. exp-sum 69.0%

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

      6. *-commutative 69.0%

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

      7. exp-to-pow 69.0%

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

      8. sub-neg 69.0%

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

      9. metadata-eval 69.0%

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

      10. exp-diff 62.1%

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

      11. *-commutative 62.1%

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

      12. exp-to-pow 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in t around 0 69.0%

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

      1. *-commutative 69.0%

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

      2. times-frac 82.8%

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

   6. Simplified 82.8%

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

   7. Taylor expanded in y around 0 49.5%

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

   8. Taylor expanded in b around 0 42.6%

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

      1. clear-num 42.6%

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

      2. inv-pow 42.6%

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

      3. distribute-lft-out 46.2%

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

      4. *-commutative 46.2%

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

   10. Applied egg-rr 46.2%

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

       1. unpow-1 46.2%

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

       2. associate-/l* 46.3%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{\frac{x}{y + y \cdot b}}}} \]

   12. Simplified 46.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{x}{y + y \cdot b}}}} \]

   if -5.5e200 < y < -1.2e11

   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. Step-by-step derivation

      1. associate-*l/ 95.6%

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

      2. *-commutative 95.6%

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

      3. +-commutative 95.6%

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

      4. associate--l+ 95.6%

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

      5. exp-sum 68.9%

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

      6. *-commutative 68.9%

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

      7. exp-to-pow 68.9%

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

      8. sub-neg 68.9%

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

      9. metadata-eval 68.9%

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

      10. exp-diff 51.1%

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

      11. *-commutative 51.1%

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

      12. exp-to-pow 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in t around 0 60.1%

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

      1. *-commutative 60.1%

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

      2. times-frac 60.2%

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

   6. Simplified 60.2%

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

   7. Taylor expanded in y around 0 52.9%

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

   8. Taylor expanded in b around 0 22.4%

      \[\leadsto \color{blue}{-1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) + \left(-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 22.4%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      2. +-commutative 22.4%

         \[\leadsto \color{blue}{\left(\frac{x}{a \cdot y} + -1 \cdot \frac{b \cdot x}{a \cdot y}\right)} + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      3. mul-1-neg 22.4%

         \[\leadsto \left(\frac{x}{a \cdot y} + \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      4. unsub-neg 22.4%

         \[\leadsto \color{blue}{\left(\frac{x}{a \cdot y} - \frac{b \cdot x}{a \cdot y}\right)} + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      5. associate-/l/ 20.4%

         \[\leadsto \left(\color{blue}{\frac{\frac{x}{y}}{a}} - \frac{b \cdot x}{a \cdot y}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      6. times-frac 20.4%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      7. mul-1-neg 20.4%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{\left(-{b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      8. distribute-rgt-neg-in 20.4%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{{b}^{2} \cdot \left(-\left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      9. distribute-rgt-out 22.7%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\color{blue}{\frac{x}{a \cdot y} \cdot \left(-1 + 0.5\right)}\right) \]

      10. metadata-eval 22.7%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\frac{x}{a \cdot y} \cdot \color{blue}{-0.5}\right) \]

      11. *-commutative 22.7%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\color{blue}{-0.5 \cdot \frac{x}{a \cdot y}}\right) \]

      12. distribute-lft-neg-in 22.7%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \color{blue}{\left(\left(--0.5\right) \cdot \frac{x}{a \cdot y}\right)} \]

      13. metadata-eval 22.7%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{x}{a \cdot y}\right) \]

      14. unpow2 22.7%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{\left(b \cdot b\right)} \cdot \left(0.5 \cdot \frac{x}{a \cdot y}\right) \]

   10. Simplified 24.9%

       \[\leadsto \color{blue}{\left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right)} \]

   11. Taylor expanded in b around inf 37.9%

       \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + 0.5 \cdot \frac{{b}^{2} \cdot x}{a \cdot y}} \]
   12. Step-by-step derivation

       1. mul-1-neg 37.9%

          \[\leadsto \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)} + 0.5 \cdot \frac{{b}^{2} \cdot x}{a \cdot y} \]

       2. times-frac 39.8%

          \[\leadsto \left(-\color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + 0.5 \cdot \frac{{b}^{2} \cdot x}{a \cdot y} \]

       3. distribute-lft-neg-out 39.8%

          \[\leadsto \color{blue}{\left(-\frac{b}{a}\right) \cdot \frac{x}{y}} + 0.5 \cdot \frac{{b}^{2} \cdot x}{a \cdot y} \]

       4. +-commutative 39.8%

          \[\leadsto \color{blue}{0.5 \cdot \frac{{b}^{2} \cdot x}{a \cdot y} + \left(-\frac{b}{a}\right) \cdot \frac{x}{y}} \]

       5. *-commutative 39.8%

          \[\leadsto 0.5 \cdot \frac{{b}^{2} \cdot x}{a \cdot y} + \color{blue}{\frac{x}{y} \cdot \left(-\frac{b}{a}\right)} \]

       6. distribute-rgt-neg-in 39.8%

          \[\leadsto 0.5 \cdot \frac{{b}^{2} \cdot x}{a \cdot y} + \color{blue}{\left(-\frac{x}{y} \cdot \frac{b}{a}\right)} \]

       7. unsub-neg 39.8%

          \[\leadsto \color{blue}{0.5 \cdot \frac{{b}^{2} \cdot x}{a \cdot y} - \frac{x}{y} \cdot \frac{b}{a}} \]

       8. associate-*r/ 39.8%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \left({b}^{2} \cdot x\right)}{a \cdot y}} - \frac{x}{y} \cdot \frac{b}{a} \]

       9. unpow2 39.8%

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

       10. associate-*l* 39.8%

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

       11. *-commutative 39.8%

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

       12. *-commutative 39.8%

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

       13. associate-*l* 39.8%

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

       14. associate-*r/ 37.8%

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

       15. times-frac 35.8%

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

       16. associate-/l* 39.9%

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

       17. associate-*r/ 39.9%

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

   13. Simplified 39.9%

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

   if -1.2e11 < y < -1.70000000000000004e-244

   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. Step-by-step derivation

      1. associate-*l/ 96.9%

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

      2. *-commutative 96.9%

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

      3. +-commutative 96.9%

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

      4. associate--l+ 96.9%

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

      5. exp-sum 93.0%

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

      6. *-commutative 93.0%

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

      7. exp-to-pow 93.9%

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

      8. sub-neg 93.9%

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

      9. metadata-eval 93.9%

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

      10. exp-diff 93.9%

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

      11. *-commutative 93.9%

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

      12. exp-to-pow 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in t around 0 81.1%

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

      1. *-commutative 81.1%

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

      2. times-frac 77.4%

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

   6. Simplified 77.4%

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

   7. Taylor expanded in y around 0 79.2%

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

   8. Taylor expanded in b around 0 21.0%

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

   9. Taylor expanded in b around 0 46.3%

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

       1. mul-1-neg 46.3%

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

       2. times-frac 44.4%

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

       3. distribute-lft-neg-out 44.4%

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

       4. associate-+r+ 44.4%

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

       5. *-commutative 44.4%

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

       6. +-commutative 44.4%

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

       7. cancel-sign-sub-inv 44.4%

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

       8. times-frac 46.3%

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

       9. *-commutative 46.3%

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

       10. div-sub 46.4%

           \[\leadsto \color{blue}{\frac{x - b \cdot x}{y \cdot a}} + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       11. *-commutative 46.4%

           \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y \cdot a} + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       12. unpow2 46.4%

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

       13. times-frac 48.4%

           \[\leadsto \frac{x - x \cdot b}{y \cdot a} + \color{blue}{\frac{b \cdot b}{a} \cdot \frac{x}{y}} \]

   11. Simplified 48.4%

       \[\leadsto \color{blue}{\frac{x - x \cdot b}{y \cdot a} + \frac{b \cdot b}{a} \cdot \frac{x}{y}} \]

   if -1.70000000000000004e-244 < y < 2.14999999999999994e-255

   1. Initial program 98.8%

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

      1. associate-*l/ 88.3%

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

      2. *-commutative 88.3%

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

      3. +-commutative 88.3%

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

      4. associate--l+ 88.3%

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

      5. exp-sum 83.0%

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

      6. *-commutative 83.0%

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

      7. exp-to-pow 84.1%

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

      8. sub-neg 84.1%

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

      9. metadata-eval 84.1%

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

      10. exp-diff 84.1%

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

      11. *-commutative 84.1%

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

      12. exp-to-pow 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in t around 0 69.7%

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

      1. *-commutative 69.7%

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

      2. times-frac 74.6%

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

   6. Simplified 74.6%

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

   7. Taylor expanded in y around 0 69.7%

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

   8. Taylor expanded in b around 0 34.2%

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

   9. Taylor expanded in y around 0 49.3%

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

   if 2.14999999999999994e-255 < y < 2.55000000000000001e-126

   1. Initial program 98.6%

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

      1. associate-*l/ 90.6%

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

      2. *-commutative 90.6%

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

      3. +-commutative 90.6%

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

      4. associate--l+ 90.6%

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

      5. exp-sum 74.6%

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

      6. *-commutative 74.6%

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

      7. exp-to-pow 75.8%

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

      8. sub-neg 75.8%

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

      9. metadata-eval 75.8%

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

      10. exp-diff 75.8%

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

      11. *-commutative 75.8%

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

      12. exp-to-pow 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in t around 0 68.0%

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

      1. *-commutative 68.0%

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

      2. times-frac 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in y around 0 68.0%

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

   8. Taylor expanded in b around 0 28.9%

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

   9. Taylor expanded in b around 0 32.8%

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

       1. mul-1-neg 32.8%

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

       2. times-frac 24.9%

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

       3. distribute-lft-neg-out 24.9%

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

       4. *-commutative 24.9%

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

       5. +-commutative 24.9%

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

       6. cancel-sign-sub-inv 24.9%

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

       7. times-frac 32.8%

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

       8. *-commutative 32.8%

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

       9. div-sub 37.1%

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

       10. associate-/r* 51.2%

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

       11. *-rgt-identity 51.2%

           \[\leadsto \frac{\frac{\color{blue}{x \cdot 1} - b \cdot x}{y}}{a} \]

       12. *-commutative 51.2%

           \[\leadsto \frac{\frac{x \cdot 1 - \color{blue}{x \cdot b}}{y}}{a} \]

       13. distribute-lft-out-- 51.2%

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

   11. Simplified 51.2%

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

   if 2.55000000000000001e-126 < y

   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. Step-by-step derivation

      1. associate-*l/ 88.0%

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

      2. *-commutative 88.0%

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

      3. +-commutative 88.0%

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

      4. associate--l+ 88.0%

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

      5. exp-sum 74.2%

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

      6. *-commutative 74.2%

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

      7. exp-to-pow 74.6%

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

      8. sub-neg 74.6%

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

      9. metadata-eval 74.6%

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

      10. exp-diff 63.1%

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

      11. *-commutative 63.1%

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

      12. exp-to-pow 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in t around 0 64.0%

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

      1. *-commutative 64.0%

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

      2. times-frac 66.3%

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

   6. Simplified 66.3%

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

   7. Taylor expanded in y around 0 51.0%

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

   8. Taylor expanded in b around 0 35.1%

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

      1. distribute-lft-out 37.5%

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

      2. *-commutative 37.5%

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

   10. Simplified 37.5%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + y \cdot b\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+200}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{x}{y + y \cdot b}}}\\ \mathbf{elif}\;y \leq -120000000000:\\ \;\;\;\;\left(b \cdot b\right) \cdot \frac{x \cdot 0.5}{y \cdot a} - \frac{x}{y \cdot \frac{a}{b}}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-244}:\\ \;\;\;\;\frac{x - x \cdot b}{y \cdot a} + \frac{x}{y} \cdot \frac{b \cdot b}{a}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-255}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y + y \cdot b\right)}\\ \end{array} \]

Alternative 14: 35.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + y \cdot b\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+200}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{x}{t_1}}}\\ \mathbf{elif}\;y \leq -1000000000000:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) - \frac{x}{y} \cdot \frac{b}{a}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-242}:\\ \;\;\;\;\frac{x - x \cdot b}{y \cdot a} + \frac{x}{y} \cdot \frac{b \cdot b}{a}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-254}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* y b))))
   (if (<= y -1.6e+200)
     (/ 1.0 (/ a (/ x t_1)))
     (if (<= y -1000000000000.0)
       (- (* (* b b) (* 0.5 (/ (/ x y) a))) (* (/ x y) (/ b a)))
       (if (<= y -3.2e-242)
         (+ (/ (- x (* x b)) (* y a)) (* (/ x y) (/ (* b b) a)))
         (if (<= y 1.8e-254)
           (/ x (* y (+ a (* a b))))
           (if (<= y 1.36e-126)
             (/ (/ (* x (- 1.0 b)) y) a)
             (/ x (* a t_1)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (y * b);
	double tmp;
	if (y <= -1.6e+200) {
		tmp = 1.0 / (a / (x / t_1));
	} else if (y <= -1000000000000.0) {
		tmp = ((b * b) * (0.5 * ((x / y) / a))) - ((x / y) * (b / a));
	} else if (y <= -3.2e-242) {
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a));
	} else if (y <= 1.8e-254) {
		tmp = x / (y * (a + (a * b)));
	} else if (y <= 1.36e-126) {
		tmp = ((x * (1.0 - b)) / y) / a;
	} else {
		tmp = x / (a * 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 = y + (y * b)
    if (y <= (-1.6d+200)) then
        tmp = 1.0d0 / (a / (x / t_1))
    else if (y <= (-1000000000000.0d0)) then
        tmp = ((b * b) * (0.5d0 * ((x / y) / a))) - ((x / y) * (b / a))
    else if (y <= (-3.2d-242)) then
        tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a))
    else if (y <= 1.8d-254) then
        tmp = x / (y * (a + (a * b)))
    else if (y <= 1.36d-126) then
        tmp = ((x * (1.0d0 - b)) / y) / a
    else
        tmp = x / (a * 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 = y + (y * b);
	double tmp;
	if (y <= -1.6e+200) {
		tmp = 1.0 / (a / (x / t_1));
	} else if (y <= -1000000000000.0) {
		tmp = ((b * b) * (0.5 * ((x / y) / a))) - ((x / y) * (b / a));
	} else if (y <= -3.2e-242) {
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a));
	} else if (y <= 1.8e-254) {
		tmp = x / (y * (a + (a * b)));
	} else if (y <= 1.36e-126) {
		tmp = ((x * (1.0 - b)) / y) / a;
	} else {
		tmp = x / (a * t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y + (y * b)
	tmp = 0
	if y <= -1.6e+200:
		tmp = 1.0 / (a / (x / t_1))
	elif y <= -1000000000000.0:
		tmp = ((b * b) * (0.5 * ((x / y) / a))) - ((x / y) * (b / a))
	elif y <= -3.2e-242:
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a))
	elif y <= 1.8e-254:
		tmp = x / (y * (a + (a * b)))
	elif y <= 1.36e-126:
		tmp = ((x * (1.0 - b)) / y) / a
	else:
		tmp = x / (a * t_1)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(y * b))
	tmp = 0.0
	if (y <= -1.6e+200)
		tmp = Float64(1.0 / Float64(a / Float64(x / t_1)));
	elseif (y <= -1000000000000.0)
		tmp = Float64(Float64(Float64(b * b) * Float64(0.5 * Float64(Float64(x / y) / a))) - Float64(Float64(x / y) * Float64(b / a)));
	elseif (y <= -3.2e-242)
		tmp = Float64(Float64(Float64(x - Float64(x * b)) / Float64(y * a)) + Float64(Float64(x / y) * Float64(Float64(b * b) / a)));
	elseif (y <= 1.8e-254)
		tmp = Float64(x / Float64(y * Float64(a + Float64(a * b))));
	elseif (y <= 1.36e-126)
		tmp = Float64(Float64(Float64(x * Float64(1.0 - b)) / y) / a);
	else
		tmp = Float64(x / Float64(a * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y + (y * b);
	tmp = 0.0;
	if (y <= -1.6e+200)
		tmp = 1.0 / (a / (x / t_1));
	elseif (y <= -1000000000000.0)
		tmp = ((b * b) * (0.5 * ((x / y) / a))) - ((x / y) * (b / a));
	elseif (y <= -3.2e-242)
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a));
	elseif (y <= 1.8e-254)
		tmp = x / (y * (a + (a * b)));
	elseif (y <= 1.36e-126)
		tmp = ((x * (1.0 - b)) / y) / a;
	else
		tmp = x / (a * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(y * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e+200], N[(1.0 / N[(a / N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1000000000000.0], N[(N[(N[(b * b), $MachinePrecision] * N[(0.5 * N[(N[(x / y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.2e-242], N[(N[(N[(x - N[(x * b), $MachinePrecision]), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e-254], N[(x / N[(y * N[(a + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.36e-126], N[(N[(N[(x * N[(1.0 - b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / a), $MachinePrecision], N[(x / N[(a * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y < -1.60000000000000016e200

   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. Step-by-step derivation

      1. associate-*l/ 93.1%

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

      2. *-commutative 93.1%

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

      3. +-commutative 93.1%

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

      4. associate--l+ 93.1%

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

      5. exp-sum 69.0%

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

      6. *-commutative 69.0%

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

      7. exp-to-pow 69.0%

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

      8. sub-neg 69.0%

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

      9. metadata-eval 69.0%

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

      10. exp-diff 62.1%

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

      11. *-commutative 62.1%

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

      12. exp-to-pow 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in t around 0 69.0%

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

      1. *-commutative 69.0%

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

      2. times-frac 82.8%

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

   6. Simplified 82.8%

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

   7. Taylor expanded in y around 0 49.5%

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

   8. Taylor expanded in b around 0 42.6%

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

      1. clear-num 42.6%

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

      2. inv-pow 42.6%

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

      3. distribute-lft-out 46.2%

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

      4. *-commutative 46.2%

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

   10. Applied egg-rr 46.2%

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

       1. unpow-1 46.2%

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

       2. associate-/l* 46.3%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{\frac{x}{y + y \cdot b}}}} \]

   12. Simplified 46.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{x}{y + y \cdot b}}}} \]

   if -1.60000000000000016e200 < y < -1e12

   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. Step-by-step derivation

      1. associate-*l/ 95.6%

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

      2. *-commutative 95.6%

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

      3. +-commutative 95.6%

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

      4. associate--l+ 95.6%

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

      5. exp-sum 68.9%

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

      6. *-commutative 68.9%

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

      7. exp-to-pow 68.9%

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

      8. sub-neg 68.9%

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

      9. metadata-eval 68.9%

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

      10. exp-diff 51.1%

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

      11. *-commutative 51.1%

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

      12. exp-to-pow 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in t around 0 60.1%

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

      1. *-commutative 60.1%

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

      2. times-frac 60.2%

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

   6. Simplified 60.2%

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

   7. Taylor expanded in y around 0 52.9%

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

   8. Taylor expanded in b around 0 22.4%

      \[\leadsto \color{blue}{-1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) + \left(-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 22.4%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      2. +-commutative 22.4%

         \[\leadsto \color{blue}{\left(\frac{x}{a \cdot y} + -1 \cdot \frac{b \cdot x}{a \cdot y}\right)} + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      3. mul-1-neg 22.4%

         \[\leadsto \left(\frac{x}{a \cdot y} + \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      4. unsub-neg 22.4%

         \[\leadsto \color{blue}{\left(\frac{x}{a \cdot y} - \frac{b \cdot x}{a \cdot y}\right)} + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      5. associate-/l/ 20.4%

         \[\leadsto \left(\color{blue}{\frac{\frac{x}{y}}{a}} - \frac{b \cdot x}{a \cdot y}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      6. times-frac 20.4%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      7. mul-1-neg 20.4%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{\left(-{b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      8. distribute-rgt-neg-in 20.4%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{{b}^{2} \cdot \left(-\left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      9. distribute-rgt-out 22.7%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\color{blue}{\frac{x}{a \cdot y} \cdot \left(-1 + 0.5\right)}\right) \]

      10. metadata-eval 22.7%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\frac{x}{a \cdot y} \cdot \color{blue}{-0.5}\right) \]

      11. *-commutative 22.7%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\color{blue}{-0.5 \cdot \frac{x}{a \cdot y}}\right) \]

      12. distribute-lft-neg-in 22.7%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \color{blue}{\left(\left(--0.5\right) \cdot \frac{x}{a \cdot y}\right)} \]

      13. metadata-eval 22.7%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{x}{a \cdot y}\right) \]

      14. unpow2 22.7%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{\left(b \cdot b\right)} \cdot \left(0.5 \cdot \frac{x}{a \cdot y}\right) \]

   10. Simplified 24.9%

       \[\leadsto \color{blue}{\left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right)} \]

   11. Taylor expanded in b around inf 35.8%

       \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]
   12. Step-by-step derivation

       1. mul-1-neg 35.8%

          \[\leadsto \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       2. times-frac 40.0%

          \[\leadsto \left(-\color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       3. distribute-lft-neg-out 40.0%

          \[\leadsto \color{blue}{\left(-\frac{b}{a}\right) \cdot \frac{x}{y}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       4. *-commutative 40.0%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot \left(-\frac{b}{a}\right)} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       5. distribute-neg-frac 40.0%

          \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{-b}{a}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

   13. Simplified 40.0%

       \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{-b}{a}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

   if -1e12 < y < -3.19999999999999999e-242

   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. Step-by-step derivation

      1. associate-*l/ 96.9%

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

      2. *-commutative 96.9%

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

      3. +-commutative 96.9%

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

      4. associate--l+ 96.9%

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

      5. exp-sum 93.0%

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

      6. *-commutative 93.0%

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

      7. exp-to-pow 93.9%

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

      8. sub-neg 93.9%

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

      9. metadata-eval 93.9%

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

      10. exp-diff 93.9%

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

      11. *-commutative 93.9%

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

      12. exp-to-pow 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in t around 0 81.1%

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

      1. *-commutative 81.1%

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

      2. times-frac 77.4%

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

   6. Simplified 77.4%

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

   7. Taylor expanded in y around 0 79.2%

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

   8. Taylor expanded in b around 0 21.0%

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

   9. Taylor expanded in b around 0 46.3%

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

       1. mul-1-neg 46.3%

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

       2. times-frac 44.4%

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

       3. distribute-lft-neg-out 44.4%

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

       4. associate-+r+ 44.4%

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

       5. *-commutative 44.4%

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

       6. +-commutative 44.4%

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

       7. cancel-sign-sub-inv 44.4%

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

       8. times-frac 46.3%

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

       9. *-commutative 46.3%

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

       10. div-sub 46.4%

           \[\leadsto \color{blue}{\frac{x - b \cdot x}{y \cdot a}} + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       11. *-commutative 46.4%

           \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y \cdot a} + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       12. unpow2 46.4%

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

       13. times-frac 48.4%

           \[\leadsto \frac{x - x \cdot b}{y \cdot a} + \color{blue}{\frac{b \cdot b}{a} \cdot \frac{x}{y}} \]

   11. Simplified 48.4%

       \[\leadsto \color{blue}{\frac{x - x \cdot b}{y \cdot a} + \frac{b \cdot b}{a} \cdot \frac{x}{y}} \]

   if -3.19999999999999999e-242 < y < 1.79999999999999992e-254

   1. Initial program 98.8%

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

      1. associate-*l/ 88.3%

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

      2. *-commutative 88.3%

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

      3. +-commutative 88.3%

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

      4. associate--l+ 88.3%

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

      5. exp-sum 83.0%

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

      6. *-commutative 83.0%

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

      7. exp-to-pow 84.1%

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

      8. sub-neg 84.1%

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

      9. metadata-eval 84.1%

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

      10. exp-diff 84.1%

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

      11. *-commutative 84.1%

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

      12. exp-to-pow 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in t around 0 69.7%

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

      1. *-commutative 69.7%

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

      2. times-frac 74.6%

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

   6. Simplified 74.6%

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

   7. Taylor expanded in y around 0 69.7%

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

   8. Taylor expanded in b around 0 34.2%

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

   9. Taylor expanded in y around 0 49.3%

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

   if 1.79999999999999992e-254 < y < 1.3599999999999999e-126

   1. Initial program 98.6%

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

      1. associate-*l/ 90.6%

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

      2. *-commutative 90.6%

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

      3. +-commutative 90.6%

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

      4. associate--l+ 90.6%

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

      5. exp-sum 74.6%

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

      6. *-commutative 74.6%

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

      7. exp-to-pow 75.8%

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

      8. sub-neg 75.8%

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

      9. metadata-eval 75.8%

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

      10. exp-diff 75.8%

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

      11. *-commutative 75.8%

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

      12. exp-to-pow 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in t around 0 68.0%

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

      1. *-commutative 68.0%

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

      2. times-frac 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in y around 0 68.0%

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

   8. Taylor expanded in b around 0 28.9%

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

   9. Taylor expanded in b around 0 32.8%

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

       1. mul-1-neg 32.8%

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

       2. times-frac 24.9%

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

       3. distribute-lft-neg-out 24.9%

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

       4. *-commutative 24.9%

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

       5. +-commutative 24.9%

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

       6. cancel-sign-sub-inv 24.9%

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

       7. times-frac 32.8%

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

       8. *-commutative 32.8%

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

       9. div-sub 37.1%

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

       10. associate-/r* 51.2%

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

       11. *-rgt-identity 51.2%

           \[\leadsto \frac{\frac{\color{blue}{x \cdot 1} - b \cdot x}{y}}{a} \]

       12. *-commutative 51.2%

           \[\leadsto \frac{\frac{x \cdot 1 - \color{blue}{x \cdot b}}{y}}{a} \]

       13. distribute-lft-out-- 51.2%

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

   11. Simplified 51.2%

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

   if 1.3599999999999999e-126 < y

   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. Step-by-step derivation

      1. associate-*l/ 88.0%

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

      2. *-commutative 88.0%

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

      3. +-commutative 88.0%

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

      4. associate--l+ 88.0%

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

      5. exp-sum 74.2%

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

      6. *-commutative 74.2%

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

      7. exp-to-pow 74.6%

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

      8. sub-neg 74.6%

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

      9. metadata-eval 74.6%

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

      10. exp-diff 63.1%

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

      11. *-commutative 63.1%

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

      12. exp-to-pow 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in t around 0 64.0%

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

      1. *-commutative 64.0%

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

      2. times-frac 66.3%

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

   6. Simplified 66.3%

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

   7. Taylor expanded in y around 0 51.0%

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

   8. Taylor expanded in b around 0 35.1%

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

      1. distribute-lft-out 37.5%

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

      2. *-commutative 37.5%

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

   10. Simplified 37.5%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + y \cdot b\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+200}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{x}{y + y \cdot b}}}\\ \mathbf{elif}\;y \leq -1000000000000:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) - \frac{x}{y} \cdot \frac{b}{a}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-242}:\\ \;\;\;\;\frac{x - x \cdot b}{y \cdot a} + \frac{x}{y} \cdot \frac{b \cdot b}{a}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-254}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y + y \cdot b\right)}\\ \end{array} \]

Alternative 15: 38.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -13500000000:\\ \;\;\;\;\frac{x}{a \cdot \frac{-y}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-126}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(\frac{b}{a} \cdot \frac{b}{y}\right)\right) + \frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y + y \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -13500000000.0)
   (+ (/ x (* a (/ (- y) b))) (* (* b b) (* 0.5 (/ (/ x y) a))))
   (if (<= y 6.2e-126)
     (+ (* x (* 0.5 (* (/ b a) (/ b y)))) (/ (/ (* x (- 1.0 b)) y) a))
     (/ x (* a (+ y (* y b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -13500000000.0) {
		tmp = (x / (a * (-y / b))) + ((b * b) * (0.5 * ((x / y) / a)));
	} else if (y <= 6.2e-126) {
		tmp = (x * (0.5 * ((b / a) * (b / y)))) + (((x * (1.0 - b)) / y) / a);
	} else {
		tmp = x / (a * (y + (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 (y <= (-13500000000.0d0)) then
        tmp = (x / (a * (-y / b))) + ((b * b) * (0.5d0 * ((x / y) / a)))
    else if (y <= 6.2d-126) then
        tmp = (x * (0.5d0 * ((b / a) * (b / y)))) + (((x * (1.0d0 - b)) / y) / a)
    else
        tmp = x / (a * (y + (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 (y <= -13500000000.0) {
		tmp = (x / (a * (-y / b))) + ((b * b) * (0.5 * ((x / y) / a)));
	} else if (y <= 6.2e-126) {
		tmp = (x * (0.5 * ((b / a) * (b / y)))) + (((x * (1.0 - b)) / y) / a);
	} else {
		tmp = x / (a * (y + (y * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -13500000000.0:
		tmp = (x / (a * (-y / b))) + ((b * b) * (0.5 * ((x / y) / a)))
	elif y <= 6.2e-126:
		tmp = (x * (0.5 * ((b / a) * (b / y)))) + (((x * (1.0 - b)) / y) / a)
	else:
		tmp = x / (a * (y + (y * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -13500000000.0)
		tmp = Float64(Float64(x / Float64(a * Float64(Float64(-y) / b))) + Float64(Float64(b * b) * Float64(0.5 * Float64(Float64(x / y) / a))));
	elseif (y <= 6.2e-126)
		tmp = Float64(Float64(x * Float64(0.5 * Float64(Float64(b / a) * Float64(b / y)))) + Float64(Float64(Float64(x * Float64(1.0 - b)) / y) / a));
	else
		tmp = Float64(x / Float64(a * Float64(y + Float64(y * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -13500000000.0)
		tmp = (x / (a * (-y / b))) + ((b * b) * (0.5 * ((x / y) / a)));
	elseif (y <= 6.2e-126)
		tmp = (x * (0.5 * ((b / a) * (b / y)))) + (((x * (1.0 - b)) / y) / a);
	else
		tmp = x / (a * (y + (y * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -13500000000.0], N[(N[(x / N[(a * N[((-y) / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 * N[(N[(x / y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e-126], N[(N[(x * N[(0.5 * N[(N[(b / a), $MachinePrecision] * N[(b / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x * N[(1.0 - b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y + N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35e10

   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. Step-by-step derivation

      1. associate-*l/ 94.6%

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

      2. *-commutative 94.6%

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

      3. +-commutative 94.6%

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

      4. associate--l+ 94.6%

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

      5. exp-sum 68.9%

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

      6. *-commutative 68.9%

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

      7. exp-to-pow 68.9%

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

      8. sub-neg 68.9%

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

      9. metadata-eval 68.9%

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

      10. exp-diff 55.4%

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

      11. *-commutative 55.4%

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

      12. exp-to-pow 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in t around 0 63.6%

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

      1. *-commutative 63.6%

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

      2. times-frac 69.0%

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

   6. Simplified 69.0%

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

   7. Taylor expanded in y around 0 51.5%

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

   8. Taylor expanded in b around 0 20.9%

      \[\leadsto \color{blue}{-1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) + \left(-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 20.9%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      2. +-commutative 20.9%

         \[\leadsto \color{blue}{\left(\frac{x}{a \cdot y} + -1 \cdot \frac{b \cdot x}{a \cdot y}\right)} + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      3. mul-1-neg 20.9%

         \[\leadsto \left(\frac{x}{a \cdot y} + \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      4. unsub-neg 20.9%

         \[\leadsto \color{blue}{\left(\frac{x}{a \cdot y} - \frac{b \cdot x}{a \cdot y}\right)} + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      5. associate-/l/ 18.8%

         \[\leadsto \left(\color{blue}{\frac{\frac{x}{y}}{a}} - \frac{b \cdot x}{a \cdot y}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      6. times-frac 21.1%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      7. mul-1-neg 21.1%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{\left(-{b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      8. distribute-rgt-neg-in 21.1%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{{b}^{2} \cdot \left(-\left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      9. distribute-rgt-out 22.5%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\color{blue}{\frac{x}{a \cdot y} \cdot \left(-1 + 0.5\right)}\right) \]

      10. metadata-eval 22.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\frac{x}{a \cdot y} \cdot \color{blue}{-0.5}\right) \]

      11. *-commutative 22.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\color{blue}{-0.5 \cdot \frac{x}{a \cdot y}}\right) \]

      12. distribute-lft-neg-in 22.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \color{blue}{\left(\left(--0.5\right) \cdot \frac{x}{a \cdot y}\right)} \]

      13. metadata-eval 22.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{x}{a \cdot y}\right) \]

      14. unpow2 22.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{\left(b \cdot b\right)} \cdot \left(0.5 \cdot \frac{x}{a \cdot y}\right) \]

   10. Simplified 23.9%

       \[\leadsto \color{blue}{\left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right)} \]

   11. Taylor expanded in b around inf 30.9%

       \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]
   12. Step-by-step derivation

       1. mul-1-neg 30.9%

          \[\leadsto \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       2. times-frac 35.6%

          \[\leadsto \left(-\color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       3. *-commutative 35.6%

          \[\leadsto \left(-\color{blue}{\frac{x}{y} \cdot \frac{b}{a}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       4. times-frac 30.9%

          \[\leadsto \left(-\color{blue}{\frac{x \cdot b}{y \cdot a}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       5. associate-/l* 36.1%

          \[\leadsto \left(-\color{blue}{\frac{x}{\frac{y \cdot a}{b}}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       6. associate-*r/ 36.1%

          \[\leadsto \left(-\frac{x}{\color{blue}{y \cdot \frac{a}{b}}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       7. *-rgt-identity 36.1%

          \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{y \cdot \frac{a}{b}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       8. times-frac 35.6%

          \[\leadsto \left(-\color{blue}{\frac{x}{y} \cdot \frac{1}{\frac{a}{b}}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       9. distribute-lft-neg-in 35.6%

          \[\leadsto \color{blue}{\left(-\frac{x}{y}\right) \cdot \frac{1}{\frac{a}{b}}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       10. *-lft-identity 35.6%

           \[\leadsto \left(-\color{blue}{1 \cdot \frac{x}{y}}\right) \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       11. metadata-eval 35.6%

           \[\leadsto \left(-\color{blue}{\frac{-1}{-1}} \cdot \frac{x}{y}\right) \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       12. times-frac 35.6%

           \[\leadsto \left(-\color{blue}{\frac{-1 \cdot x}{-1 \cdot y}}\right) \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       13. neg-mul-1 35.6%

           \[\leadsto \left(-\frac{\color{blue}{-x}}{-1 \cdot y}\right) \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       14. neg-mul-1 35.6%

           \[\leadsto \left(-\frac{-x}{\color{blue}{-y}}\right) \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       15. distribute-frac-neg 35.6%

           \[\leadsto \color{blue}{\frac{-\left(-x\right)}{-y}} \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       16. remove-double-neg 35.6%

           \[\leadsto \frac{\color{blue}{x}}{-y} \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       17. times-frac 36.1%

           \[\leadsto \color{blue}{\frac{x \cdot 1}{\left(-y\right) \cdot \frac{a}{b}}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       18. *-rgt-identity 36.1%

           \[\leadsto \frac{\color{blue}{x}}{\left(-y\right) \cdot \frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       19. associate-*r/ 36.1%

           \[\leadsto \frac{x}{\color{blue}{\frac{\left(-y\right) \cdot a}{b}}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       20. distribute-lft-neg-in 36.1%

           \[\leadsto \frac{x}{\frac{\color{blue}{-y \cdot a}}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       21. distribute-rgt-neg-in 36.1%

           \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot \left(-a\right)}}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       22. *-commutative 36.1%

           \[\leadsto \frac{x}{\frac{\color{blue}{\left(-a\right) \cdot y}}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       23. *-lft-identity 36.1%

           \[\leadsto \frac{x}{\frac{\left(-a\right) \cdot y}{\color{blue}{1 \cdot b}}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       24. times-frac 38.4%

           \[\leadsto \frac{x}{\color{blue}{\frac{-a}{1} \cdot \frac{y}{b}}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

   13. Simplified 38.4%

       \[\leadsto \color{blue}{\frac{x}{\left(-a\right) \cdot \frac{y}{b}}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

   if -1.35e10 < y < 6.2000000000000003e-126

   1. Initial program 97.5%

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

      1. associate-*l/ 93.5%

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

      2. *-commutative 93.5%

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

      3. +-commutative 93.5%

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

      4. associate--l+ 93.5%

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

      5. exp-sum 86.2%

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

      6. *-commutative 86.2%

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

      7. exp-to-pow 87.2%

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

      8. sub-neg 87.2%

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

      9. metadata-eval 87.2%

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

      10. exp-diff 87.2%

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

      11. *-commutative 87.2%

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

      12. exp-to-pow 87.2%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in t around 0 75.4%

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

      1. *-commutative 75.4%

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

      2. times-frac 76.1%

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

   6. Simplified 76.1%

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

   7. Taylor expanded in y around 0 74.4%

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

   8. Taylor expanded in b around 0 23.1%

      \[\leadsto \color{blue}{-1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) + \left(-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 23.1%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      2. +-commutative 23.1%

         \[\leadsto \color{blue}{\left(\frac{x}{a \cdot y} + -1 \cdot \frac{b \cdot x}{a \cdot y}\right)} + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      3. mul-1-neg 23.1%

         \[\leadsto \left(\frac{x}{a \cdot y} + \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      4. unsub-neg 23.1%

         \[\leadsto \color{blue}{\left(\frac{x}{a \cdot y} - \frac{b \cdot x}{a \cdot y}\right)} + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      5. associate-/l/ 24.7%

         \[\leadsto \left(\color{blue}{\frac{\frac{x}{y}}{a}} - \frac{b \cdot x}{a \cdot y}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      6. times-frac 23.7%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      7. mul-1-neg 23.7%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{\left(-{b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      8. distribute-rgt-neg-in 23.7%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{{b}^{2} \cdot \left(-\left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      9. distribute-rgt-out 37.5%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\color{blue}{\frac{x}{a \cdot y} \cdot \left(-1 + 0.5\right)}\right) \]

      10. metadata-eval 37.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\frac{x}{a \cdot y} \cdot \color{blue}{-0.5}\right) \]

      11. *-commutative 37.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\color{blue}{-0.5 \cdot \frac{x}{a \cdot y}}\right) \]

      12. distribute-lft-neg-in 37.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \color{blue}{\left(\left(--0.5\right) \cdot \frac{x}{a \cdot y}\right)} \]

      13. metadata-eval 37.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{x}{a \cdot y}\right) \]

      14. unpow2 37.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{\left(b \cdot b\right)} \cdot \left(0.5 \cdot \frac{x}{a \cdot y}\right) \]

   10. Simplified 38.6%

       \[\leadsto \color{blue}{\left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right)} \]

   11. Taylor expanded in x around 0 37.5%

       \[\leadsto \color{blue}{x \cdot \left(\left(0.5 \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{a \cdot y}\right) - \frac{b}{a \cdot y}\right)} \]

   13. Simplified 48.0%

       \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(\frac{b}{a} \cdot \frac{b}{y}\right)\right) + \frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}} \]

   if 6.2000000000000003e-126 < y

   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. Step-by-step derivation

      1. associate-*l/ 88.0%

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

      2. *-commutative 88.0%

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

      3. +-commutative 88.0%

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

      4. associate--l+ 88.0%

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

      5. exp-sum 74.2%

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

      6. *-commutative 74.2%

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

      7. exp-to-pow 74.6%

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

      8. sub-neg 74.6%

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

      9. metadata-eval 74.6%

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

      10. exp-diff 63.1%

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

      11. *-commutative 63.1%

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

      12. exp-to-pow 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in t around 0 64.0%

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

      1. *-commutative 64.0%

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

      2. times-frac 66.3%

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

   6. Simplified 66.3%

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

   7. Taylor expanded in y around 0 51.0%

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

   8. Taylor expanded in b around 0 35.1%

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

      1. distribute-lft-out 37.5%

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

      2. *-commutative 37.5%

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

   10. Simplified 37.5%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13500000000:\\ \;\;\;\;\frac{x}{a \cdot \frac{-y}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-126}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(\frac{b}{a} \cdot \frac{b}{y}\right)\right) + \frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y + y \cdot b\right)}\\ \end{array} \]

Alternative 16: 35.8% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -122000000000:\\ \;\;\;\;\frac{x}{a \cdot \frac{-y}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right)\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-244}:\\ \;\;\;\;\frac{x - x \cdot b}{y \cdot a} + \frac{x}{y} \cdot \frac{b \cdot b}{a}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-256}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y + y \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -122000000000.0)
   (+ (/ x (* a (/ (- y) b))) (* (* b b) (* 0.5 (/ (/ x y) a))))
   (if (<= y -9.5e-244)
     (+ (/ (- x (* x b)) (* y a)) (* (/ x y) (/ (* b b) a)))
     (if (<= y 5.2e-256)
       (/ x (* y (+ a (* a b))))
       (if (<= y 4.5e-126)
         (/ (/ (* x (- 1.0 b)) y) a)
         (/ x (* a (+ y (* y b)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -122000000000.0) {
		tmp = (x / (a * (-y / b))) + ((b * b) * (0.5 * ((x / y) / a)));
	} else if (y <= -9.5e-244) {
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a));
	} else if (y <= 5.2e-256) {
		tmp = x / (y * (a + (a * b)));
	} else if (y <= 4.5e-126) {
		tmp = ((x * (1.0 - b)) / y) / a;
	} else {
		tmp = x / (a * (y + (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 (y <= (-122000000000.0d0)) then
        tmp = (x / (a * (-y / b))) + ((b * b) * (0.5d0 * ((x / y) / a)))
    else if (y <= (-9.5d-244)) then
        tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a))
    else if (y <= 5.2d-256) then
        tmp = x / (y * (a + (a * b)))
    else if (y <= 4.5d-126) then
        tmp = ((x * (1.0d0 - b)) / y) / a
    else
        tmp = x / (a * (y + (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 (y <= -122000000000.0) {
		tmp = (x / (a * (-y / b))) + ((b * b) * (0.5 * ((x / y) / a)));
	} else if (y <= -9.5e-244) {
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a));
	} else if (y <= 5.2e-256) {
		tmp = x / (y * (a + (a * b)));
	} else if (y <= 4.5e-126) {
		tmp = ((x * (1.0 - b)) / y) / a;
	} else {
		tmp = x / (a * (y + (y * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -122000000000.0:
		tmp = (x / (a * (-y / b))) + ((b * b) * (0.5 * ((x / y) / a)))
	elif y <= -9.5e-244:
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a))
	elif y <= 5.2e-256:
		tmp = x / (y * (a + (a * b)))
	elif y <= 4.5e-126:
		tmp = ((x * (1.0 - b)) / y) / a
	else:
		tmp = x / (a * (y + (y * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -122000000000.0)
		tmp = Float64(Float64(x / Float64(a * Float64(Float64(-y) / b))) + Float64(Float64(b * b) * Float64(0.5 * Float64(Float64(x / y) / a))));
	elseif (y <= -9.5e-244)
		tmp = Float64(Float64(Float64(x - Float64(x * b)) / Float64(y * a)) + Float64(Float64(x / y) * Float64(Float64(b * b) / a)));
	elseif (y <= 5.2e-256)
		tmp = Float64(x / Float64(y * Float64(a + Float64(a * b))));
	elseif (y <= 4.5e-126)
		tmp = Float64(Float64(Float64(x * Float64(1.0 - b)) / y) / a);
	else
		tmp = Float64(x / Float64(a * Float64(y + Float64(y * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -122000000000.0)
		tmp = (x / (a * (-y / b))) + ((b * b) * (0.5 * ((x / y) / a)));
	elseif (y <= -9.5e-244)
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a));
	elseif (y <= 5.2e-256)
		tmp = x / (y * (a + (a * b)));
	elseif (y <= 4.5e-126)
		tmp = ((x * (1.0 - b)) / y) / a;
	else
		tmp = x / (a * (y + (y * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -122000000000.0], N[(N[(x / N[(a * N[((-y) / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 * N[(N[(x / y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -9.5e-244], N[(N[(N[(x - N[(x * b), $MachinePrecision]), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e-256], N[(x / N[(y * N[(a + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e-126], N[(N[(N[(x * N[(1.0 - b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / a), $MachinePrecision], N[(x / N[(a * N[(y + N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.22e11

   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. Step-by-step derivation

      1. associate-*l/ 94.6%

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

      2. *-commutative 94.6%

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

      3. +-commutative 94.6%

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

      4. associate--l+ 94.6%

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

      5. exp-sum 68.9%

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

      6. *-commutative 68.9%

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

      7. exp-to-pow 68.9%

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

      8. sub-neg 68.9%

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

      9. metadata-eval 68.9%

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

      10. exp-diff 55.4%

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

      11. *-commutative 55.4%

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

      12. exp-to-pow 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in t around 0 63.6%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 63.6%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 69.0%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 51.5%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 20.9%

      \[\leadsto \color{blue}{-1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) + \left(-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 20.9%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      2. +-commutative 20.9%

         \[\leadsto \color{blue}{\left(\frac{x}{a \cdot y} + -1 \cdot \frac{b \cdot x}{a \cdot y}\right)} + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      3. mul-1-neg 20.9%

         \[\leadsto \left(\frac{x}{a \cdot y} + \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      4. unsub-neg 20.9%

         \[\leadsto \color{blue}{\left(\frac{x}{a \cdot y} - \frac{b \cdot x}{a \cdot y}\right)} + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      5. associate-/l/ 18.8%

         \[\leadsto \left(\color{blue}{\frac{\frac{x}{y}}{a}} - \frac{b \cdot x}{a \cdot y}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      6. times-frac 21.1%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + -1 \cdot \left({b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right) \]

      7. mul-1-neg 21.1%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{\left(-{b}^{2} \cdot \left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      8. distribute-rgt-neg-in 21.1%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{{b}^{2} \cdot \left(-\left(-1 \cdot \frac{x}{a \cdot y} + 0.5 \cdot \frac{x}{a \cdot y}\right)\right)} \]

      9. distribute-rgt-out 22.5%

         \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\color{blue}{\frac{x}{a \cdot y} \cdot \left(-1 + 0.5\right)}\right) \]

      10. metadata-eval 22.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\frac{x}{a \cdot y} \cdot \color{blue}{-0.5}\right) \]

      11. *-commutative 22.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(-\color{blue}{-0.5 \cdot \frac{x}{a \cdot y}}\right) \]

      12. distribute-lft-neg-in 22.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \color{blue}{\left(\left(--0.5\right) \cdot \frac{x}{a \cdot y}\right)} \]

      13. metadata-eval 22.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + {b}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{x}{a \cdot y}\right) \]

      14. unpow2 22.5%

          \[\leadsto \left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \color{blue}{\left(b \cdot b\right)} \cdot \left(0.5 \cdot \frac{x}{a \cdot y}\right) \]

   10. Simplified 23.9%

       \[\leadsto \color{blue}{\left(\frac{\frac{x}{y}}{a} - \frac{b}{a} \cdot \frac{x}{y}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right)} \]

   11. Taylor expanded in b around inf 30.9%

       \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]
   12. Step-by-step derivation

       1. mul-1-neg 30.9%

          \[\leadsto \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       2. times-frac 35.6%

          \[\leadsto \left(-\color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       3. *-commutative 35.6%

          \[\leadsto \left(-\color{blue}{\frac{x}{y} \cdot \frac{b}{a}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       4. times-frac 30.9%

          \[\leadsto \left(-\color{blue}{\frac{x \cdot b}{y \cdot a}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       5. associate-/l* 36.1%

          \[\leadsto \left(-\color{blue}{\frac{x}{\frac{y \cdot a}{b}}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       6. associate-*r/ 36.1%

          \[\leadsto \left(-\frac{x}{\color{blue}{y \cdot \frac{a}{b}}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       7. *-rgt-identity 36.1%

          \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{y \cdot \frac{a}{b}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       8. times-frac 35.6%

          \[\leadsto \left(-\color{blue}{\frac{x}{y} \cdot \frac{1}{\frac{a}{b}}}\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       9. distribute-lft-neg-in 35.6%

          \[\leadsto \color{blue}{\left(-\frac{x}{y}\right) \cdot \frac{1}{\frac{a}{b}}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       10. *-lft-identity 35.6%

           \[\leadsto \left(-\color{blue}{1 \cdot \frac{x}{y}}\right) \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       11. metadata-eval 35.6%

           \[\leadsto \left(-\color{blue}{\frac{-1}{-1}} \cdot \frac{x}{y}\right) \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       12. times-frac 35.6%

           \[\leadsto \left(-\color{blue}{\frac{-1 \cdot x}{-1 \cdot y}}\right) \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       13. neg-mul-1 35.6%

           \[\leadsto \left(-\frac{\color{blue}{-x}}{-1 \cdot y}\right) \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       14. neg-mul-1 35.6%

           \[\leadsto \left(-\frac{-x}{\color{blue}{-y}}\right) \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       15. distribute-frac-neg 35.6%

           \[\leadsto \color{blue}{\frac{-\left(-x\right)}{-y}} \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       16. remove-double-neg 35.6%

           \[\leadsto \frac{\color{blue}{x}}{-y} \cdot \frac{1}{\frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       17. times-frac 36.1%

           \[\leadsto \color{blue}{\frac{x \cdot 1}{\left(-y\right) \cdot \frac{a}{b}}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       18. *-rgt-identity 36.1%

           \[\leadsto \frac{\color{blue}{x}}{\left(-y\right) \cdot \frac{a}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       19. associate-*r/ 36.1%

           \[\leadsto \frac{x}{\color{blue}{\frac{\left(-y\right) \cdot a}{b}}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       20. distribute-lft-neg-in 36.1%

           \[\leadsto \frac{x}{\frac{\color{blue}{-y \cdot a}}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       21. distribute-rgt-neg-in 36.1%

           \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot \left(-a\right)}}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       22. *-commutative 36.1%

           \[\leadsto \frac{x}{\frac{\color{blue}{\left(-a\right) \cdot y}}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       23. *-lft-identity 36.1%

           \[\leadsto \frac{x}{\frac{\left(-a\right) \cdot y}{\color{blue}{1 \cdot b}}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

       24. times-frac 38.4%

           \[\leadsto \frac{x}{\color{blue}{\frac{-a}{1} \cdot \frac{y}{b}}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

   13. Simplified 38.4%

       \[\leadsto \color{blue}{\frac{x}{\left(-a\right) \cdot \frac{y}{b}}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right) \]

   if -1.22e11 < y < -9.4999999999999995e-244

   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. Step-by-step derivation

      1. associate-*l/ 96.9%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 96.9%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 96.9%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 96.9%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 93.0%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 93.0%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 93.9%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 93.9%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 93.9%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 93.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 93.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 93.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 93.9%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 81.1%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 81.1%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 79.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 21.0%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in b around 0 46.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + \left(\frac{x}{a \cdot y} + \frac{{b}^{2} \cdot x}{a \cdot y}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 46.3%

          \[\leadsto \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)} + \left(\frac{x}{a \cdot y} + \frac{{b}^{2} \cdot x}{a \cdot y}\right) \]

       2. times-frac 44.4%

          \[\leadsto \left(-\color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + \left(\frac{x}{a \cdot y} + \frac{{b}^{2} \cdot x}{a \cdot y}\right) \]

       3. distribute-lft-neg-out 44.4%

          \[\leadsto \color{blue}{\left(-\frac{b}{a}\right) \cdot \frac{x}{y}} + \left(\frac{x}{a \cdot y} + \frac{{b}^{2} \cdot x}{a \cdot y}\right) \]

       4. associate-+r+ 44.4%

          \[\leadsto \color{blue}{\left(\left(-\frac{b}{a}\right) \cdot \frac{x}{y} + \frac{x}{a \cdot y}\right) + \frac{{b}^{2} \cdot x}{a \cdot y}} \]

       5. *-commutative 44.4%

          \[\leadsto \left(\left(-\frac{b}{a}\right) \cdot \frac{x}{y} + \frac{x}{\color{blue}{y \cdot a}}\right) + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       6. +-commutative 44.4%

          \[\leadsto \color{blue}{\left(\frac{x}{y \cdot a} + \left(-\frac{b}{a}\right) \cdot \frac{x}{y}\right)} + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       7. cancel-sign-sub-inv 44.4%

          \[\leadsto \color{blue}{\left(\frac{x}{y \cdot a} - \frac{b}{a} \cdot \frac{x}{y}\right)} + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       8. times-frac 46.3%

          \[\leadsto \left(\frac{x}{y \cdot a} - \color{blue}{\frac{b \cdot x}{a \cdot y}}\right) + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       9. *-commutative 46.3%

          \[\leadsto \left(\frac{x}{y \cdot a} - \frac{b \cdot x}{\color{blue}{y \cdot a}}\right) + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       10. div-sub 46.4%

           \[\leadsto \color{blue}{\frac{x - b \cdot x}{y \cdot a}} + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       11. *-commutative 46.4%

           \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y \cdot a} + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       12. unpow2 46.4%

           \[\leadsto \frac{x - x \cdot b}{y \cdot a} + \frac{\color{blue}{\left(b \cdot b\right)} \cdot x}{a \cdot y} \]

       13. times-frac 48.4%

           \[\leadsto \frac{x - x \cdot b}{y \cdot a} + \color{blue}{\frac{b \cdot b}{a} \cdot \frac{x}{y}} \]

   11. Simplified 48.4%

       \[\leadsto \color{blue}{\frac{x - x \cdot b}{y \cdot a} + \frac{b \cdot b}{a} \cdot \frac{x}{y}} \]

   if -9.4999999999999995e-244 < y < 5.2000000000000002e-256

   1. Initial program 98.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 88.3%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 88.3%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 88.3%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 88.3%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 83.0%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 83.0%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 84.1%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 84.1%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 84.1%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 84.1%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 84.1%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 84.1%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 84.1%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 69.7%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 69.7%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 74.6%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 69.7%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 34.2%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in y around 0 49.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(a + a \cdot b\right)}} \]

   if 5.2000000000000002e-256 < y < 4.50000000000000025e-126

   1. Initial program 98.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 90.6%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 90.6%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 90.6%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 90.6%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 74.6%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 74.6%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 75.8%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 75.8%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 75.8%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 75.8%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 75.8%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 75.8%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 68.0%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 68.0%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 74.5%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 74.5%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 68.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 28.9%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in b around 0 32.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. mul-1-neg 32.8%

          \[\leadsto \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)} + \frac{x}{a \cdot y} \]

       2. times-frac 24.9%

          \[\leadsto \left(-\color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + \frac{x}{a \cdot y} \]

       3. distribute-lft-neg-out 24.9%

          \[\leadsto \color{blue}{\left(-\frac{b}{a}\right) \cdot \frac{x}{y}} + \frac{x}{a \cdot y} \]

       4. *-commutative 24.9%

          \[\leadsto \left(-\frac{b}{a}\right) \cdot \frac{x}{y} + \frac{x}{\color{blue}{y \cdot a}} \]

       5. +-commutative 24.9%

          \[\leadsto \color{blue}{\frac{x}{y \cdot a} + \left(-\frac{b}{a}\right) \cdot \frac{x}{y}} \]

       6. cancel-sign-sub-inv 24.9%

          \[\leadsto \color{blue}{\frac{x}{y \cdot a} - \frac{b}{a} \cdot \frac{x}{y}} \]

       7. times-frac 32.8%

          \[\leadsto \frac{x}{y \cdot a} - \color{blue}{\frac{b \cdot x}{a \cdot y}} \]

       8. *-commutative 32.8%

          \[\leadsto \frac{x}{y \cdot a} - \frac{b \cdot x}{\color{blue}{y \cdot a}} \]

       9. div-sub 37.1%

          \[\leadsto \color{blue}{\frac{x - b \cdot x}{y \cdot a}} \]

       10. associate-/r* 51.2%

           \[\leadsto \color{blue}{\frac{\frac{x - b \cdot x}{y}}{a}} \]

       11. *-rgt-identity 51.2%

           \[\leadsto \frac{\frac{\color{blue}{x \cdot 1} - b \cdot x}{y}}{a} \]

       12. *-commutative 51.2%

           \[\leadsto \frac{\frac{x \cdot 1 - \color{blue}{x \cdot b}}{y}}{a} \]

       13. distribute-lft-out-- 51.2%

           \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(1 - b\right)}}{y}}{a} \]

   11. Simplified 51.2%

       \[\leadsto \color{blue}{\frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}} \]

   if 4.50000000000000025e-126 < y

   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. Step-by-step derivation

      1. associate-*l/ 88.0%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 88.0%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 88.0%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 88.0%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 74.2%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 74.2%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 74.6%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 74.6%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 74.6%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 63.1%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 63.1%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 63.1%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 63.1%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 64.0%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 64.0%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 66.3%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 66.3%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 51.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 35.1%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 37.5%

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

      2. *-commutative 37.5%

         \[\leadsto \frac{x}{a \cdot \left(y + \color{blue}{y \cdot b}\right)} \]

   10. Simplified 37.5%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + y \cdot b\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -122000000000:\\ \;\;\;\;\frac{x}{a \cdot \frac{-y}{b}} + \left(b \cdot b\right) \cdot \left(0.5 \cdot \frac{\frac{x}{y}}{a}\right)\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-244}:\\ \;\;\;\;\frac{x - x \cdot b}{y \cdot a} + \frac{x}{y} \cdot \frac{b \cdot b}{a}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-256}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y + y \cdot b\right)}\\ \end{array} \]

Alternative 17: 43.1% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - b\right)}{y}}{a} + \frac{b \cdot b}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -9e-137)
   (+ (/ (/ (* x (- 1.0 b)) y) a) (/ (* b b) (* a (/ y x))))
   (/ x (* y (+ a (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -9e-137) {
		tmp = (((x * (1.0 - b)) / y) / a) + ((b * b) / (a * (y / x)));
	} else {
		tmp = x / (y * (a + (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 <= (-9d-137)) then
        tmp = (((x * (1.0d0 - b)) / y) / a) + ((b * b) / (a * (y / x)))
    else
        tmp = x / (y * (a + (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 <= -9e-137) {
		tmp = (((x * (1.0 - b)) / y) / a) + ((b * b) / (a * (y / x)));
	} else {
		tmp = x / (y * (a + (a * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -9e-137:
		tmp = (((x * (1.0 - b)) / y) / a) + ((b * b) / (a * (y / x)))
	else:
		tmp = x / (y * (a + (a * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -9e-137)
		tmp = Float64(Float64(Float64(Float64(x * Float64(1.0 - b)) / y) / a) + Float64(Float64(b * b) / Float64(a * Float64(y / x))));
	else
		tmp = Float64(x / Float64(y * Float64(a + Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -9e-137)
		tmp = (((x * (1.0 - b)) / y) / a) + ((b * b) / (a * (y / x)));
	else
		tmp = x / (y * (a + (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -9e-137], N[(N[(N[(N[(x * N[(1.0 - b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] / N[(a * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(a + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.9999999999999994e-137

   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. Step-by-step derivation

      1. associate-*l/ 93.6%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 93.6%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 93.6%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 93.6%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 79.3%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 79.3%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 79.6%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 79.6%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 79.6%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 66.5%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 66.5%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 66.5%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 66.5%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 71.8%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 71.8%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 76.1%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 76.1%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 71.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 16.6%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in b around 0 45.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + \left(\frac{x}{a \cdot y} + \frac{{b}^{2} \cdot x}{a \cdot y}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 45.8%

          \[\leadsto \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)} + \left(\frac{x}{a \cdot y} + \frac{{b}^{2} \cdot x}{a \cdot y}\right) \]

       2. times-frac 45.8%

          \[\leadsto \left(-\color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + \left(\frac{x}{a \cdot y} + \frac{{b}^{2} \cdot x}{a \cdot y}\right) \]

       3. distribute-lft-neg-out 45.8%

          \[\leadsto \color{blue}{\left(-\frac{b}{a}\right) \cdot \frac{x}{y}} + \left(\frac{x}{a \cdot y} + \frac{{b}^{2} \cdot x}{a \cdot y}\right) \]

       4. +-commutative 45.8%

          \[\leadsto \color{blue}{\left(\frac{x}{a \cdot y} + \frac{{b}^{2} \cdot x}{a \cdot y}\right) + \left(-\frac{b}{a}\right) \cdot \frac{x}{y}} \]

       5. *-commutative 45.8%

          \[\leadsto \left(\frac{x}{\color{blue}{y \cdot a}} + \frac{{b}^{2} \cdot x}{a \cdot y}\right) + \left(-\frac{b}{a}\right) \cdot \frac{x}{y} \]

       6. +-commutative 45.8%

          \[\leadsto \color{blue}{\left(\frac{{b}^{2} \cdot x}{a \cdot y} + \frac{x}{y \cdot a}\right)} + \left(-\frac{b}{a}\right) \cdot \frac{x}{y} \]

       7. associate-+l+ 45.8%

          \[\leadsto \color{blue}{\frac{{b}^{2} \cdot x}{a \cdot y} + \left(\frac{x}{y \cdot a} + \left(-\frac{b}{a}\right) \cdot \frac{x}{y}\right)} \]

       8. unpow2 45.8%

          \[\leadsto \frac{\color{blue}{\left(b \cdot b\right)} \cdot x}{a \cdot y} + \left(\frac{x}{y \cdot a} + \left(-\frac{b}{a}\right) \cdot \frac{x}{y}\right) \]

       9. associate-/l* 43.6%

          \[\leadsto \color{blue}{\frac{b \cdot b}{\frac{a \cdot y}{x}}} + \left(\frac{x}{y \cdot a} + \left(-\frac{b}{a}\right) \cdot \frac{x}{y}\right) \]

       10. associate-*r/ 43.5%

           \[\leadsto \frac{b \cdot b}{\color{blue}{a \cdot \frac{y}{x}}} + \left(\frac{x}{y \cdot a} + \left(-\frac{b}{a}\right) \cdot \frac{x}{y}\right) \]

       11. cancel-sign-sub-inv 43.5%

           \[\leadsto \frac{b \cdot b}{a \cdot \frac{y}{x}} + \color{blue}{\left(\frac{x}{y \cdot a} - \frac{b}{a} \cdot \frac{x}{y}\right)} \]

       12. times-frac 43.5%

           \[\leadsto \frac{b \cdot b}{a \cdot \frac{y}{x}} + \left(\frac{x}{y \cdot a} - \color{blue}{\frac{b \cdot x}{a \cdot y}}\right) \]

       13. *-commutative 43.5%

           \[\leadsto \frac{b \cdot b}{a \cdot \frac{y}{x}} + \left(\frac{x}{y \cdot a} - \frac{b \cdot x}{\color{blue}{y \cdot a}}\right) \]

       14. div-sub 43.5%

           \[\leadsto \frac{b \cdot b}{a \cdot \frac{y}{x}} + \color{blue}{\frac{x - b \cdot x}{y \cdot a}} \]

   11. Simplified 45.5%

       \[\leadsto \color{blue}{\frac{b \cdot b}{a \cdot \frac{y}{x}} + \frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}} \]

   if -8.9999999999999994e-137 < b

   1. Initial program 98.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 91.2%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 91.2%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 91.2%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 91.2%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 76.1%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 76.1%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 76.6%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 76.6%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 76.6%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 71.4%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 71.4%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 71.4%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 71.4%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 66.3%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 68.1%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 54.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 34.0%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in y around 0 38.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(a + a \cdot b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - b\right)}{y}}{a} + \frac{b \cdot b}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \end{array} \]

Alternative 18: 43.7% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -65000:\\ \;\;\;\;\frac{x - x \cdot b}{y \cdot a} + \frac{x}{y} \cdot \frac{b \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -65000.0)
   (+ (/ (- x (* x b)) (* y a)) (* (/ x y) (/ (* b b) a)))
   (/ x (* y (+ a (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -65000.0) {
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a));
	} else {
		tmp = x / (y * (a + (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 <= (-65000.0d0)) then
        tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a))
    else
        tmp = x / (y * (a + (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 <= -65000.0) {
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a));
	} else {
		tmp = x / (y * (a + (a * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -65000.0:
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a))
	else:
		tmp = x / (y * (a + (a * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -65000.0)
		tmp = Float64(Float64(Float64(x - Float64(x * b)) / Float64(y * a)) + Float64(Float64(x / y) * Float64(Float64(b * b) / a)));
	else
		tmp = Float64(x / Float64(y * Float64(a + Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -65000.0)
		tmp = ((x - (x * b)) / (y * a)) + ((x / y) * ((b * b) / a));
	else
		tmp = x / (y * (a + (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -65000.0], N[(N[(N[(x - N[(x * b), $MachinePrecision]), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(a + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -65000

   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. Step-by-step derivation

      1. associate-*l/ 96.7%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 96.7%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 96.7%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 96.7%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 82.0%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 82.0%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 82.0%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 82.0%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 82.0%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 63.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 63.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 63.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 75.5%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 75.5%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 75.5%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 85.5%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 13.7%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in b around 0 52.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + \left(\frac{x}{a \cdot y} + \frac{{b}^{2} \cdot x}{a \cdot y}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 52.2%

          \[\leadsto \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)} + \left(\frac{x}{a \cdot y} + \frac{{b}^{2} \cdot x}{a \cdot y}\right) \]

       2. times-frac 52.2%

          \[\leadsto \left(-\color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + \left(\frac{x}{a \cdot y} + \frac{{b}^{2} \cdot x}{a \cdot y}\right) \]

       3. distribute-lft-neg-out 52.2%

          \[\leadsto \color{blue}{\left(-\frac{b}{a}\right) \cdot \frac{x}{y}} + \left(\frac{x}{a \cdot y} + \frac{{b}^{2} \cdot x}{a \cdot y}\right) \]

       4. associate-+r+ 52.2%

          \[\leadsto \color{blue}{\left(\left(-\frac{b}{a}\right) \cdot \frac{x}{y} + \frac{x}{a \cdot y}\right) + \frac{{b}^{2} \cdot x}{a \cdot y}} \]

       5. *-commutative 52.2%

          \[\leadsto \left(\left(-\frac{b}{a}\right) \cdot \frac{x}{y} + \frac{x}{\color{blue}{y \cdot a}}\right) + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       6. +-commutative 52.2%

          \[\leadsto \color{blue}{\left(\frac{x}{y \cdot a} + \left(-\frac{b}{a}\right) \cdot \frac{x}{y}\right)} + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       7. cancel-sign-sub-inv 52.2%

          \[\leadsto \color{blue}{\left(\frac{x}{y \cdot a} - \frac{b}{a} \cdot \frac{x}{y}\right)} + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       8. times-frac 52.2%

          \[\leadsto \left(\frac{x}{y \cdot a} - \color{blue}{\frac{b \cdot x}{a \cdot y}}\right) + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       9. *-commutative 52.2%

          \[\leadsto \left(\frac{x}{y \cdot a} - \frac{b \cdot x}{\color{blue}{y \cdot a}}\right) + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       10. div-sub 52.2%

           \[\leadsto \color{blue}{\frac{x - b \cdot x}{y \cdot a}} + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       11. *-commutative 52.2%

           \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y \cdot a} + \frac{{b}^{2} \cdot x}{a \cdot y} \]

       12. unpow2 52.2%

           \[\leadsto \frac{x - x \cdot b}{y \cdot a} + \frac{\color{blue}{\left(b \cdot b\right)} \cdot x}{a \cdot y} \]

       13. times-frac 52.2%

           \[\leadsto \frac{x - x \cdot b}{y \cdot a} + \color{blue}{\frac{b \cdot b}{a} \cdot \frac{x}{y}} \]

   11. Simplified 52.2%

       \[\leadsto \color{blue}{\frac{x - x \cdot b}{y \cdot a} + \frac{b \cdot b}{a} \cdot \frac{x}{y}} \]

   if -65000 < b

   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. Step-by-step derivation

      1. associate-*l/ 90.5%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 90.5%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 90.5%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 90.5%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 75.6%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 75.6%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 76.3%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 76.3%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 76.3%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 71.6%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 71.6%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 71.6%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 65.8%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 65.8%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 69.2%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 51.8%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 32.9%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in y around 0 36.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(a + a \cdot b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -65000:\\ \;\;\;\;\frac{x - x \cdot b}{y \cdot a} + \frac{x}{y} \cdot \frac{b \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \end{array} \]

Alternative 19: 39.5% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-268}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{x}{a}}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1e+62)
   (/ (/ (* x (- 1.0 b)) y) a)
   (if (<= b 1.02e-268)
     (/ (/ x a) y)
     (if (<= b 4.1e-86)
       (/ x (* y a))
       (if (<= b 6.8e+95) (/ (/ x a) (* y b)) (/ x (* y (* a b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1e+62) {
		tmp = ((x * (1.0 - b)) / y) / a;
	} else if (b <= 1.02e-268) {
		tmp = (x / a) / y;
	} else if (b <= 4.1e-86) {
		tmp = x / (y * a);
	} else if (b <= 6.8e+95) {
		tmp = (x / a) / (y * b);
	} 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 <= (-1d+62)) then
        tmp = ((x * (1.0d0 - b)) / y) / a
    else if (b <= 1.02d-268) then
        tmp = (x / a) / y
    else if (b <= 4.1d-86) then
        tmp = x / (y * a)
    else if (b <= 6.8d+95) then
        tmp = (x / a) / (y * b)
    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 <= -1e+62) {
		tmp = ((x * (1.0 - b)) / y) / a;
	} else if (b <= 1.02e-268) {
		tmp = (x / a) / y;
	} else if (b <= 4.1e-86) {
		tmp = x / (y * a);
	} else if (b <= 6.8e+95) {
		tmp = (x / a) / (y * b);
	} else {
		tmp = x / (y * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1e+62:
		tmp = ((x * (1.0 - b)) / y) / a
	elif b <= 1.02e-268:
		tmp = (x / a) / y
	elif b <= 4.1e-86:
		tmp = x / (y * a)
	elif b <= 6.8e+95:
		tmp = (x / a) / (y * b)
	else:
		tmp = x / (y * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1e+62)
		tmp = Float64(Float64(Float64(x * Float64(1.0 - b)) / y) / a);
	elseif (b <= 1.02e-268)
		tmp = Float64(Float64(x / a) / y);
	elseif (b <= 4.1e-86)
		tmp = Float64(x / Float64(y * a));
	elseif (b <= 6.8e+95)
		tmp = Float64(Float64(x / a) / Float64(y * b));
	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 <= -1e+62)
		tmp = ((x * (1.0 - b)) / y) / a;
	elseif (b <= 1.02e-268)
		tmp = (x / a) / y;
	elseif (b <= 4.1e-86)
		tmp = x / (y * a);
	elseif (b <= 6.8e+95)
		tmp = (x / a) / (y * b);
	else
		tmp = x / (y * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1e+62], N[(N[(N[(x * N[(1.0 - b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.02e-268], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 4.1e-86], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e+95], N[(N[(x / a), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.00000000000000004e62

   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. Step-by-step derivation

      1. associate-*l/ 96.1%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 96.1%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 96.1%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 96.1%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 78.4%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 78.4%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 78.4%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 78.4%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 78.4%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 56.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 56.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 56.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 56.9%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 70.7%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 70.7%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 70.7%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 70.7%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 82.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 16.2%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in b around 0 41.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. mul-1-neg 41.5%

          \[\leadsto \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)} + \frac{x}{a \cdot y} \]

       2. times-frac 39.5%

          \[\leadsto \left(-\color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + \frac{x}{a \cdot y} \]

       3. distribute-lft-neg-out 39.5%

          \[\leadsto \color{blue}{\left(-\frac{b}{a}\right) \cdot \frac{x}{y}} + \frac{x}{a \cdot y} \]

       4. *-commutative 39.5%

          \[\leadsto \left(-\frac{b}{a}\right) \cdot \frac{x}{y} + \frac{x}{\color{blue}{y \cdot a}} \]

       5. +-commutative 39.5%

          \[\leadsto \color{blue}{\frac{x}{y \cdot a} + \left(-\frac{b}{a}\right) \cdot \frac{x}{y}} \]

       6. cancel-sign-sub-inv 39.5%

          \[\leadsto \color{blue}{\frac{x}{y \cdot a} - \frac{b}{a} \cdot \frac{x}{y}} \]

       7. times-frac 41.5%

          \[\leadsto \frac{x}{y \cdot a} - \color{blue}{\frac{b \cdot x}{a \cdot y}} \]

       8. *-commutative 41.5%

          \[\leadsto \frac{x}{y \cdot a} - \frac{b \cdot x}{\color{blue}{y \cdot a}} \]

       9. div-sub 41.5%

          \[\leadsto \color{blue}{\frac{x - b \cdot x}{y \cdot a}} \]

       10. associate-/r* 46.9%

           \[\leadsto \color{blue}{\frac{\frac{x - b \cdot x}{y}}{a}} \]

       11. *-rgt-identity 46.9%

           \[\leadsto \frac{\frac{\color{blue}{x \cdot 1} - b \cdot x}{y}}{a} \]

       12. *-commutative 46.9%

           \[\leadsto \frac{\frac{x \cdot 1 - \color{blue}{x \cdot b}}{y}}{a} \]

       13. distribute-lft-out-- 46.9%

           \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(1 - b\right)}}{y}}{a} \]

   11. Simplified 46.9%

       \[\leadsto \color{blue}{\frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}} \]

   if -1.00000000000000004e62 < b < 1.0200000000000001e-268

   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. Step-by-step derivation

      1. associate-/l* 96.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}} \]

      2. fma-def 96.8%

         \[\leadsto \frac{x}{\frac{y}{e^{\color{blue}{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right)} - b}}} \]

      3. sub-neg 96.8%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \color{blue}{\left(t + \left(-1\right)\right)} \cdot \log a\right) - b}}} \]

      4. metadata-eval 96.8%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + \color{blue}{-1}\right) \cdot \log a\right) - b}}} \]

   3. Simplified 96.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + -1\right) \cdot \log a\right) - b}}}} \]

   4. Taylor expanded in t around 0 79.0%

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

   5. Taylor expanded in b around 0 70.0%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}}{y} \]
   6. Step-by-step derivation

      1. +-commutative 70.0%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 70.0%

         \[\leadsto \frac{x \cdot e^{y \cdot \log z + \color{blue}{\left(-\log a\right)}}}{y} \]

      3. sub-neg 70.0%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z - \log a}}}{y} \]

      4. exp-diff 70.0%

         \[\leadsto \frac{x \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y} \]

      5. *-commutative 70.0%

         \[\leadsto \frac{x \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y} \]

      6. exp-to-pow 70.0%

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y} \]

      7. rem-exp-log 70.9%

         \[\leadsto \frac{x \cdot \frac{{z}^{y}}{\color{blue}{a}}}{y} \]

      8. *-commutative 70.9%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y}}{a} \cdot x}}{y} \]

   7. Simplified 70.9%

      \[\leadsto \frac{\color{blue}{\frac{{z}^{y}}{a} \cdot x}}{y} \]

   8. Taylor expanded in y around 0 31.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

   if 1.0200000000000001e-268 < b < 4.09999999999999979e-86

   1. Initial program 96.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 92.1%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 92.1%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 92.1%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 92.1%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 78.2%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 78.2%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 79.0%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 79.0%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 79.0%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 79.0%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 79.0%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 79.0%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 79.0%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 65.9%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 65.9%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 65.9%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 41.4%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 41.4%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]
   9. Step-by-step derivation

      1. *-commutative 41.4%

         \[\leadsto \frac{x}{\color{blue}{y \cdot a}} \]

   10. Simplified 41.4%

       \[\leadsto \frac{x}{\color{blue}{y \cdot a}} \]

   if 4.09999999999999979e-86 < b < 6.80000000000000043e95

   1. Initial program 99.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 99.4%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 99.4%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 99.4%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 85.9%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 85.9%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 86.5%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 86.5%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 86.5%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 81.1%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 81.1%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 81.1%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 81.1%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 65.9%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 65.9%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 73.8%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 61.1%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 29.9%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in b around inf 32.5%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. associate-/r* 35.1%

          \[\leadsto \color{blue}{\frac{\frac{x}{a}}{b \cdot y}} \]

       2. *-commutative 35.1%

          \[\leadsto \frac{\frac{x}{a}}{\color{blue}{y \cdot b}} \]

   11. Simplified 35.1%

       \[\leadsto \color{blue}{\frac{\frac{x}{a}}{y \cdot b}} \]

   if 6.80000000000000043e95 < 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. Step-by-step derivation

      1. associate-*l/ 83.7%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 83.7%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 83.7%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 83.7%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 63.3%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 63.3%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 63.3%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 63.3%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 63.3%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 49.0%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 49.0%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 49.0%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 49.0%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 67.5%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 67.5%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 67.5%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 81.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 42.5%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in b around inf 42.5%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 42.5%

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot b\right)}} \]

       2. *-commutative 42.5%

          \[\leadsto \frac{x}{\color{blue}{\left(y \cdot b\right) \cdot a}} \]

       3. associate-*l* 48.2%

          \[\leadsto \frac{x}{\color{blue}{y \cdot \left(b \cdot a\right)}} \]

       4. *-commutative 48.2%

          \[\leadsto \frac{x}{y \cdot \color{blue}{\left(a \cdot b\right)}} \]

   11. Simplified 48.2%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \left(a \cdot b\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - b\right)}{y}}{a}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-268}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{x}{a}}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 20: 39.3% accurate, 24.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+62}:\\ \;\;\;\;\frac{x - x \cdot b}{y \cdot a}\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.02e+62)
   (/ (- x (* x b)) (* y a))
   (if (<= b -3.5e-132) (/ (/ x a) y) (/ x (* y (+ a (* a b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.02e+62) {
		tmp = (x - (x * b)) / (y * a);
	} else if (b <= -3.5e-132) {
		tmp = (x / a) / y;
	} else {
		tmp = x / (y * (a + (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 <= (-1.02d+62)) then
        tmp = (x - (x * b)) / (y * a)
    else if (b <= (-3.5d-132)) then
        tmp = (x / a) / y
    else
        tmp = x / (y * (a + (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 <= -1.02e+62) {
		tmp = (x - (x * b)) / (y * a);
	} else if (b <= -3.5e-132) {
		tmp = (x / a) / y;
	} else {
		tmp = x / (y * (a + (a * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1.02e+62:
		tmp = (x - (x * b)) / (y * a)
	elif b <= -3.5e-132:
		tmp = (x / a) / y
	else:
		tmp = x / (y * (a + (a * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.02e+62)
		tmp = Float64(Float64(x - Float64(x * b)) / Float64(y * a));
	elseif (b <= -3.5e-132)
		tmp = Float64(Float64(x / a) / y);
	else
		tmp = Float64(x / Float64(y * Float64(a + Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.02e+62)
		tmp = (x - (x * b)) / (y * a);
	elseif (b <= -3.5e-132)
		tmp = (x / a) / y;
	else
		tmp = x / (y * (a + (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.02e+62], N[(N[(x - N[(x * b), $MachinePrecision]), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3.5e-132], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(y * N[(a + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.02000000000000002e62

   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. Step-by-step derivation

      1. associate-*l/ 96.1%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 96.1%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 96.1%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 96.1%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 78.4%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 78.4%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 78.4%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 78.4%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 78.4%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 56.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 56.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 56.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 56.9%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 70.7%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 70.7%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 70.7%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 70.7%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 82.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 16.2%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in b around 0 41.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. mul-1-neg 41.5%

          \[\leadsto \color{blue}{\left(-\frac{b \cdot x}{a \cdot y}\right)} + \frac{x}{a \cdot y} \]

       2. times-frac 39.5%

          \[\leadsto \left(-\color{blue}{\frac{b}{a} \cdot \frac{x}{y}}\right) + \frac{x}{a \cdot y} \]

       3. distribute-lft-neg-out 39.5%

          \[\leadsto \color{blue}{\left(-\frac{b}{a}\right) \cdot \frac{x}{y}} + \frac{x}{a \cdot y} \]

       4. *-commutative 39.5%

          \[\leadsto \left(-\frac{b}{a}\right) \cdot \frac{x}{y} + \frac{x}{\color{blue}{y \cdot a}} \]

       5. +-commutative 39.5%

          \[\leadsto \color{blue}{\frac{x}{y \cdot a} + \left(-\frac{b}{a}\right) \cdot \frac{x}{y}} \]

       6. cancel-sign-sub-inv 39.5%

          \[\leadsto \color{blue}{\frac{x}{y \cdot a} - \frac{b}{a} \cdot \frac{x}{y}} \]

       7. times-frac 41.5%

          \[\leadsto \frac{x}{y \cdot a} - \color{blue}{\frac{b \cdot x}{a \cdot y}} \]

       8. *-commutative 41.5%

          \[\leadsto \frac{x}{y \cdot a} - \frac{b \cdot x}{\color{blue}{y \cdot a}} \]

       9. div-sub 41.5%

          \[\leadsto \color{blue}{\frac{x - b \cdot x}{y \cdot a}} \]

       10. *-commutative 41.5%

           \[\leadsto \frac{x - \color{blue}{x \cdot b}}{y \cdot a} \]

   11. Simplified 41.5%

       \[\leadsto \color{blue}{\frac{x - x \cdot b}{y \cdot a}} \]

   if -1.02000000000000002e62 < b < -3.5e-132

   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. Step-by-step derivation

      1. associate-/l* 94.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}} \]

      2. fma-def 94.2%

         \[\leadsto \frac{x}{\frac{y}{e^{\color{blue}{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right)} - b}}} \]

      3. sub-neg 94.2%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \color{blue}{\left(t + \left(-1\right)\right)} \cdot \log a\right) - b}}} \]

      4. metadata-eval 94.2%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + \color{blue}{-1}\right) \cdot \log a\right) - b}}} \]

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + -1\right) \cdot \log a\right) - b}}}} \]

   4. Taylor expanded in t around 0 92.4%

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

   5. Taylor expanded in b around 0 71.0%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}}{y} \]
   6. Step-by-step derivation

      1. +-commutative 71.0%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 71.0%

         \[\leadsto \frac{x \cdot e^{y \cdot \log z + \color{blue}{\left(-\log a\right)}}}{y} \]

      3. sub-neg 71.0%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z - \log a}}}{y} \]

      4. exp-diff 71.0%

         \[\leadsto \frac{x \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y} \]

      5. *-commutative 71.0%

         \[\leadsto \frac{x \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y} \]

      6. exp-to-pow 71.0%

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y} \]

      7. rem-exp-log 72.0%

         \[\leadsto \frac{x \cdot \frac{{z}^{y}}{\color{blue}{a}}}{y} \]

      8. *-commutative 72.0%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y}}{a} \cdot x}}{y} \]

   7. Simplified 72.0%

      \[\leadsto \frac{\color{blue}{\frac{{z}^{y}}{a} \cdot x}}{y} \]

   8. Taylor expanded in y around 0 35.3%

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

   if -3.5e-132 < b

   1. Initial program 98.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 91.2%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 91.2%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 91.2%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 91.2%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 76.2%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 76.2%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 76.8%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 76.8%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 76.8%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 71.6%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 71.6%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 71.6%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 65.9%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 65.9%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 67.7%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 67.7%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 54.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 33.8%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in y around 0 37.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(a + a \cdot b\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+62}:\\ \;\;\;\;\frac{x - x \cdot b}{y \cdot a}\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \end{array} \]

Alternative 21: 36.0% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.35e-125) (/ 1.0 (* a (/ y x))) (/ x (* y (+ a (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.35e-125) {
		tmp = 1.0 / (a * (y / x));
	} else {
		tmp = x / (y * (a + (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-125)) then
        tmp = 1.0d0 / (a * (y / x))
    else
        tmp = x / (y * (a + (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-125) {
		tmp = 1.0 / (a * (y / x));
	} else {
		tmp = x / (y * (a + (a * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1.35e-125:
		tmp = 1.0 / (a * (y / x))
	else:
		tmp = x / (y * (a + (a * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.35e-125)
		tmp = Float64(1.0 / Float64(a * Float64(y / x)));
	else
		tmp = Float64(x / Float64(y * Float64(a + 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-125)
		tmp = 1.0 / (a * (y / x));
	else
		tmp = x / (y * (a + (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.35e-125], N[(1.0 / N[(a * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(a + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3499999999999999e-125

   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. Step-by-step derivation

      1. associate-/l* 97.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}} \]

      2. fma-def 97.8%

         \[\leadsto \frac{x}{\frac{y}{e^{\color{blue}{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right)} - b}}} \]

      3. sub-neg 97.8%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \color{blue}{\left(t + \left(-1\right)\right)} \cdot \log a\right) - b}}} \]

      4. metadata-eval 97.8%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + \color{blue}{-1}\right) \cdot \log a\right) - b}}} \]

   3. Simplified 97.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + -1\right) \cdot \log a\right) - b}}}} \]

   4. Taylor expanded in t around 0 93.5%

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

   5. Taylor expanded in b around 0 58.6%

      \[\leadsto \color{blue}{\frac{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}{y}} \]
   6. Step-by-step derivation

      1. +-commutative 58.6%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 58.6%

         \[\leadsto \frac{x \cdot e^{y \cdot \log z + \color{blue}{\left(-\log a\right)}}}{y} \]

      3. sub-neg 58.6%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z - \log a}}}{y} \]

      4. exp-diff 58.6%

         \[\leadsto \frac{x \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y} \]

      5. *-commutative 58.6%

         \[\leadsto \frac{x \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y} \]

      6. exp-to-pow 58.6%

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y} \]

      7. rem-exp-log 58.9%

         \[\leadsto \frac{x \cdot \frac{{z}^{y}}{\color{blue}{a}}}{y} \]

      8. *-commutative 58.9%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y}}{a} \cdot x}}{y} \]

      9. associate-*l/ 58.9%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y} \cdot x}{a}}}{y} \]

      10. *-commutative 58.9%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot {z}^{y}}}{a}}{y} \]

      11. associate-/r* 49.6%

          \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot y}} \]

      12. *-commutative 49.6%

          \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot y} \]

      13. times-frac 55.2%

          \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y}} \]

   7. Simplified 55.2%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y}} \]

   8. Taylor expanded in y around 0 26.3%

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   9. Step-by-step derivation

      1. *-commutative 26.3%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{a}} \]

      2. clear-num 27.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \cdot \frac{1}{a} \]

      3. frac-times 27.5%

         \[\leadsto \color{blue}{\frac{1 \cdot 1}{\frac{y}{x} \cdot a}} \]

      4. metadata-eval 27.5%

         \[\leadsto \frac{\color{blue}{1}}{\frac{y}{x} \cdot a} \]

   10. Applied egg-rr 27.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x} \cdot a}} \]

   if -1.3499999999999999e-125 < b

   1. Initial program 98.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 91.2%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 91.2%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 91.2%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 91.2%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 76.2%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 76.2%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 76.8%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 76.8%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 76.8%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 71.6%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 71.6%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 71.6%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 65.9%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 65.9%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 67.7%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 67.7%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 54.0%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 33.8%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in y around 0 37.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(a + a \cdot b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \end{array} \]

Alternative 22: 30.9% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 3.5e-98) (/ 1.0 (* a (/ y x))) (/ x (* y a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 3.5e-98) {
		tmp = 1.0 / (a * (y / x));
	} else {
		tmp = x / (y * a);
	}
	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 (z <= 3.5d-98) then
        tmp = 1.0d0 / (a * (y / x))
    else
        tmp = x / (y * a)
    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 (z <= 3.5e-98) {
		tmp = 1.0 / (a * (y / x));
	} else {
		tmp = x / (y * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= 3.5e-98:
		tmp = 1.0 / (a * (y / x))
	else:
		tmp = x / (y * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 3.5e-98)
		tmp = Float64(1.0 / Float64(a * Float64(y / x)));
	else
		tmp = Float64(x / Float64(y * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= 3.5e-98)
		tmp = 1.0 / (a * (y / x));
	else
		tmp = x / (y * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 3.5e-98], N[(1.0 / N[(a * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.5000000000000002e-98

   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. Step-by-step derivation

      1. associate-/l* 96.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}} \]

      2. fma-def 96.8%

         \[\leadsto \frac{x}{\frac{y}{e^{\color{blue}{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right)} - b}}} \]

      3. sub-neg 96.8%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \color{blue}{\left(t + \left(-1\right)\right)} \cdot \log a\right) - b}}} \]

      4. metadata-eval 96.8%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + \color{blue}{-1}\right) \cdot \log a\right) - b}}} \]

   3. Simplified 96.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + -1\right) \cdot \log a\right) - b}}}} \]

   4. Taylor expanded in t around 0 86.1%

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

   5. Taylor expanded in b around 0 69.2%

      \[\leadsto \color{blue}{\frac{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}{y}} \]
   6. Step-by-step derivation

      1. +-commutative 69.2%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 69.2%

         \[\leadsto \frac{x \cdot e^{y \cdot \log z + \color{blue}{\left(-\log a\right)}}}{y} \]

      3. sub-neg 69.2%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z - \log a}}}{y} \]

      4. exp-diff 69.2%

         \[\leadsto \frac{x \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y} \]

      5. *-commutative 69.2%

         \[\leadsto \frac{x \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y} \]

      6. exp-to-pow 69.2%

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y} \]

      7. rem-exp-log 70.1%

         \[\leadsto \frac{x \cdot \frac{{z}^{y}}{\color{blue}{a}}}{y} \]

      8. *-commutative 70.1%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y}}{a} \cdot x}}{y} \]

      9. associate-*l/ 70.1%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y} \cdot x}{a}}}{y} \]

      10. *-commutative 70.1%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot {z}^{y}}}{a}}{y} \]

      11. associate-/r* 57.0%

          \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot y}} \]

      12. *-commutative 57.0%

          \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot y} \]

      13. times-frac 70.0%

          \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y}} \]

   7. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y}} \]

   8. Taylor expanded in y around 0 35.6%

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   9. Step-by-step derivation

      1. *-commutative 35.6%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{a}} \]

      2. clear-num 36.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \cdot \frac{1}{a} \]

      3. frac-times 36.9%

         \[\leadsto \color{blue}{\frac{1 \cdot 1}{\frac{y}{x} \cdot a}} \]

      4. metadata-eval 36.9%

         \[\leadsto \frac{\color{blue}{1}}{\frac{y}{x} \cdot a} \]

   10. Applied egg-rr 36.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x} \cdot a}} \]

   if 3.5000000000000002e-98 < z

   1. Initial program 98.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 90.7%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 90.7%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 90.7%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 90.7%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 75.2%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 75.2%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 75.6%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 75.6%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 75.6%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 67.8%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 67.8%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 67.8%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 67.8%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 68.0%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 68.0%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 68.4%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 68.4%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 60.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 28.3%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]
   9. Step-by-step derivation

      1. *-commutative 28.3%

         \[\leadsto \frac{x}{\color{blue}{y \cdot a}} \]

   10. Simplified 28.3%

       \[\leadsto \frac{x}{\color{blue}{y \cdot a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]

Alternative 23: 31.0% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y \cdot a}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 5e-98) (/ 1.0 (* a (/ y x))) (/ 1.0 (/ (* y a) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 5e-98) {
		tmp = 1.0 / (a * (y / x));
	} else {
		tmp = 1.0 / ((y * a) / x);
	}
	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 (z <= 5d-98) then
        tmp = 1.0d0 / (a * (y / x))
    else
        tmp = 1.0d0 / ((y * a) / x)
    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 (z <= 5e-98) {
		tmp = 1.0 / (a * (y / x));
	} else {
		tmp = 1.0 / ((y * a) / x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= 5e-98:
		tmp = 1.0 / (a * (y / x))
	else:
		tmp = 1.0 / ((y * a) / x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 5e-98)
		tmp = Float64(1.0 / Float64(a * Float64(y / x)));
	else
		tmp = Float64(1.0 / Float64(Float64(y * a) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= 5e-98)
		tmp = 1.0 / (a * (y / x));
	else
		tmp = 1.0 / ((y * a) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 5e-98], N[(1.0 / N[(a * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y * a), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.00000000000000018e-98

   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. Step-by-step derivation

      1. associate-/l* 96.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}} \]

      2. fma-def 96.8%

         \[\leadsto \frac{x}{\frac{y}{e^{\color{blue}{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right)} - b}}} \]

      3. sub-neg 96.8%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \color{blue}{\left(t + \left(-1\right)\right)} \cdot \log a\right) - b}}} \]

      4. metadata-eval 96.8%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + \color{blue}{-1}\right) \cdot \log a\right) - b}}} \]

   3. Simplified 96.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + -1\right) \cdot \log a\right) - b}}}} \]

   4. Taylor expanded in t around 0 86.1%

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

   5. Taylor expanded in b around 0 69.2%

      \[\leadsto \color{blue}{\frac{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}{y}} \]
   6. Step-by-step derivation

      1. +-commutative 69.2%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 69.2%

         \[\leadsto \frac{x \cdot e^{y \cdot \log z + \color{blue}{\left(-\log a\right)}}}{y} \]

      3. sub-neg 69.2%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z - \log a}}}{y} \]

      4. exp-diff 69.2%

         \[\leadsto \frac{x \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y} \]

      5. *-commutative 69.2%

         \[\leadsto \frac{x \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y} \]

      6. exp-to-pow 69.2%

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y} \]

      7. rem-exp-log 70.1%

         \[\leadsto \frac{x \cdot \frac{{z}^{y}}{\color{blue}{a}}}{y} \]

      8. *-commutative 70.1%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y}}{a} \cdot x}}{y} \]

      9. associate-*l/ 70.1%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y} \cdot x}{a}}}{y} \]

      10. *-commutative 70.1%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot {z}^{y}}}{a}}{y} \]

      11. associate-/r* 57.0%

          \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot y}} \]

      12. *-commutative 57.0%

          \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot y} \]

      13. times-frac 70.0%

          \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y}} \]

   7. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y}} \]

   8. Taylor expanded in y around 0 35.6%

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   9. Step-by-step derivation

      1. *-commutative 35.6%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{a}} \]

      2. clear-num 36.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \cdot \frac{1}{a} \]

      3. frac-times 36.9%

         \[\leadsto \color{blue}{\frac{1 \cdot 1}{\frac{y}{x} \cdot a}} \]

      4. metadata-eval 36.9%

         \[\leadsto \frac{\color{blue}{1}}{\frac{y}{x} \cdot a} \]

   10. Applied egg-rr 36.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x} \cdot a}} \]

   if 5.00000000000000018e-98 < z

   1. Initial program 98.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}} \]

      2. fma-def 99.2%

         \[\leadsto \frac{x}{\frac{y}{e^{\color{blue}{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right)} - b}}} \]

      3. sub-neg 99.2%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \color{blue}{\left(t + \left(-1\right)\right)} \cdot \log a\right) - b}}} \]

      4. metadata-eval 99.2%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + \color{blue}{-1}\right) \cdot \log a\right) - b}}} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + -1\right) \cdot \log a\right) - b}}}} \]

   4. Taylor expanded in t around 0 81.5%

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

   5. Taylor expanded in b around 0 57.3%

      \[\leadsto \color{blue}{\frac{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}{y}} \]
   6. Step-by-step derivation

      1. +-commutative 57.3%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 57.3%

         \[\leadsto \frac{x \cdot e^{y \cdot \log z + \color{blue}{\left(-\log a\right)}}}{y} \]

      3. sub-neg 57.3%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z - \log a}}}{y} \]

      4. exp-diff 57.3%

         \[\leadsto \frac{x \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y} \]

      5. *-commutative 57.3%

         \[\leadsto \frac{x \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y} \]

      6. exp-to-pow 57.3%

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y} \]

      7. rem-exp-log 57.5%

         \[\leadsto \frac{x \cdot \frac{{z}^{y}}{\color{blue}{a}}}{y} \]

      8. *-commutative 57.5%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y}}{a} \cdot x}}{y} \]

      9. associate-*l/ 57.5%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y} \cdot x}{a}}}{y} \]

      10. *-commutative 57.5%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot {z}^{y}}}{a}}{y} \]

      11. associate-/r* 51.6%

          \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot y}} \]

      12. *-commutative 51.6%

          \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot y} \]

      13. times-frac 49.9%

          \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y}} \]

   7. Simplified 49.9%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y}} \]

   8. Taylor expanded in y around 0 24.0%

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   9. Step-by-step derivation

      1. associate-*l/ 24.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y}}{a}} \]

      2. *-un-lft-identity 24.0%

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{a} \]

      3. associate-/r* 28.3%

         \[\leadsto \color{blue}{\frac{x}{y \cdot a}} \]

      4. clear-num 28.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot a}{x}}} \]

   10. Applied egg-rr 28.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot a}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y \cdot a}{x}}\\ \end{array} \]

Alternative 24: 34.7% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 1.95e-79) (/ 1.0 (* a (/ y x))) (/ x (* a (* y b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1.95e-79) {
		tmp = 1.0 / (a * (y / x));
	} 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 <= 1.95d-79) then
        tmp = 1.0d0 / (a * (y / x))
    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 <= 1.95e-79) {
		tmp = 1.0 / (a * (y / x));
	} else {
		tmp = x / (a * (y * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= 1.95e-79:
		tmp = 1.0 / (a * (y / x))
	else:
		tmp = x / (a * (y * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 1.95e-79)
		tmp = Float64(1.0 / Float64(a * Float64(y / x)));
	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 <= 1.95e-79)
		tmp = 1.0 / (a * (y / x));
	else
		tmp = x / (a * (y * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 1.95e-79], N[(1.0 / N[(a * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.95000000000000003e-79

   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. Step-by-step derivation

      1. associate-/l* 98.4%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}} \]

      2. fma-def 98.4%

         \[\leadsto \frac{x}{\frac{y}{e^{\color{blue}{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right)} - b}}} \]

      3. sub-neg 98.4%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \color{blue}{\left(t + \left(-1\right)\right)} \cdot \log a\right) - b}}} \]

      4. metadata-eval 98.4%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + \color{blue}{-1}\right) \cdot \log a\right) - b}}} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + -1\right) \cdot \log a\right) - b}}}} \]

   4. Taylor expanded in t around 0 81.3%

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

   5. Taylor expanded in b around 0 64.4%

      \[\leadsto \color{blue}{\frac{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}{y}} \]
   6. Step-by-step derivation

      1. +-commutative 64.4%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 64.4%

         \[\leadsto \frac{x \cdot e^{y \cdot \log z + \color{blue}{\left(-\log a\right)}}}{y} \]

      3. sub-neg 64.4%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z - \log a}}}{y} \]

      4. exp-diff 64.4%

         \[\leadsto \frac{x \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y} \]

      5. *-commutative 64.4%

         \[\leadsto \frac{x \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y} \]

      6. exp-to-pow 64.4%

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y} \]

      7. rem-exp-log 65.0%

         \[\leadsto \frac{x \cdot \frac{{z}^{y}}{\color{blue}{a}}}{y} \]

      8. *-commutative 65.0%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y}}{a} \cdot x}}{y} \]

      9. associate-*l/ 65.0%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y} \cdot x}{a}}}{y} \]

      10. *-commutative 65.0%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot {z}^{y}}}{a}}{y} \]

      11. associate-/r* 56.9%

          \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot y}} \]

      12. *-commutative 56.9%

          \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot y} \]

      13. times-frac 60.8%

          \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y}} \]

   7. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y}} \]

   8. Taylor expanded in y around 0 29.5%

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   9. Step-by-step derivation

      1. *-commutative 29.5%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{a}} \]

      2. clear-num 30.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \cdot \frac{1}{a} \]

      3. frac-times 30.1%

         \[\leadsto \color{blue}{\frac{1 \cdot 1}{\frac{y}{x} \cdot a}} \]

      4. metadata-eval 30.1%

         \[\leadsto \frac{\color{blue}{1}}{\frac{y}{x} \cdot a} \]

   10. Applied egg-rr 30.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x} \cdot a}} \]

   if 1.95000000000000003e-79 < b

   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. Step-by-step derivation

      1. associate-*l/ 90.2%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 90.2%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 90.2%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 90.2%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 72.4%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 72.4%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 72.6%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 72.6%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 72.6%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 61.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 61.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 61.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 68.3%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 68.3%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 69.5%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 69.5%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 74.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 37.8%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in b around inf 37.9%

      \[\leadsto \frac{x}{\color{blue}{a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 37.9%

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot b\right)}} \]

   11. Simplified 37.9%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 25: 35.4% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 2.8e-27) (/ 1.0 (* a (/ y x))) (/ x (* y (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 2.8e-27) {
		tmp = 1.0 / (a * (y / x));
	} 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 <= 2.8d-27) then
        tmp = 1.0d0 / (a * (y / x))
    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 <= 2.8e-27) {
		tmp = 1.0 / (a * (y / x));
	} else {
		tmp = x / (y * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= 2.8e-27:
		tmp = 1.0 / (a * (y / x))
	else:
		tmp = x / (y * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 2.8e-27)
		tmp = Float64(1.0 / Float64(a * Float64(y / x)));
	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 <= 2.8e-27)
		tmp = 1.0 / (a * (y / x));
	else
		tmp = x / (y * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 2.8e-27], N[(1.0 / N[(a * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8e-27

   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. Step-by-step derivation

      1. associate-/l* 97.9%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}} \]

      2. fma-def 97.9%

         \[\leadsto \frac{x}{\frac{y}{e^{\color{blue}{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right)} - b}}} \]

      3. sub-neg 97.9%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \color{blue}{\left(t + \left(-1\right)\right)} \cdot \log a\right) - b}}} \]

      4. metadata-eval 97.9%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + \color{blue}{-1}\right) \cdot \log a\right) - b}}} \]

   3. Simplified 97.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + -1\right) \cdot \log a\right) - b}}}} \]

   4. Taylor expanded in t around 0 80.2%

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

   5. Taylor expanded in b around 0 64.4%

      \[\leadsto \color{blue}{\frac{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}{y}} \]
   6. Step-by-step derivation

      1. +-commutative 64.4%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 64.4%

         \[\leadsto \frac{x \cdot e^{y \cdot \log z + \color{blue}{\left(-\log a\right)}}}{y} \]

      3. sub-neg 64.4%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z - \log a}}}{y} \]

      4. exp-diff 64.4%

         \[\leadsto \frac{x \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y} \]

      5. *-commutative 64.4%

         \[\leadsto \frac{x \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y} \]

      6. exp-to-pow 64.4%

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y} \]

      7. rem-exp-log 65.0%

         \[\leadsto \frac{x \cdot \frac{{z}^{y}}{\color{blue}{a}}}{y} \]

      8. *-commutative 65.0%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y}}{a} \cdot x}}{y} \]

      9. associate-*l/ 65.0%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y} \cdot x}{a}}}{y} \]

      10. *-commutative 65.0%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot {z}^{y}}}{a}}{y} \]

      11. associate-/r* 56.4%

          \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot y}} \]

      12. *-commutative 56.4%

          \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot y} \]

      13. times-frac 61.1%

          \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y}} \]

   7. Simplified 61.1%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y}} \]

   8. Taylor expanded in y around 0 29.6%

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   9. Step-by-step derivation

      1. *-commutative 29.6%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{a}} \]

      2. clear-num 30.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \cdot \frac{1}{a} \]

      3. frac-times 30.1%

         \[\leadsto \color{blue}{\frac{1 \cdot 1}{\frac{y}{x} \cdot a}} \]

      4. metadata-eval 30.1%

         \[\leadsto \frac{\color{blue}{1}}{\frac{y}{x} \cdot a} \]

   10. Applied egg-rr 30.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x} \cdot a}} \]

   if 2.8e-27 < 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. Step-by-step derivation

      1. associate-*l/ 89.0%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 89.0%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 89.0%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 89.0%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 69.9%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 69.9%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 69.9%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 69.9%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 69.9%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 57.5%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 57.5%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 57.5%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 57.5%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 71.4%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 71.4%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 70.1%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 81.1%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 38.8%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

   9. Taylor expanded in b around inf 38.8%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 38.8%

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot b\right)}} \]

       2. *-commutative 38.8%

          \[\leadsto \frac{x}{\color{blue}{\left(y \cdot b\right) \cdot a}} \]

       3. associate-*l* 42.7%

          \[\leadsto \frac{x}{\color{blue}{y \cdot \left(b \cdot a\right)}} \]

       4. *-commutative 42.7%

          \[\leadsto \frac{x}{y \cdot \color{blue}{\left(a \cdot b\right)}} \]

   11. Simplified 42.7%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \left(a \cdot b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 26: 31.2% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 1.6e+54) (/ (/ x a) y) (/ x (* y a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 1.6e+54) {
		tmp = (x / a) / y;
	} else {
		tmp = x / (y * a);
	}
	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 (z <= 1.6d+54) then
        tmp = (x / a) / y
    else
        tmp = x / (y * a)
    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 (z <= 1.6e+54) {
		tmp = (x / a) / y;
	} else {
		tmp = x / (y * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= 1.6e+54:
		tmp = (x / a) / y
	else:
		tmp = x / (y * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 1.6e+54)
		tmp = Float64(Float64(x / a) / y);
	else
		tmp = Float64(x / Float64(y * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= 1.6e+54)
		tmp = (x / a) / y;
	else
		tmp = x / (y * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 1.6e+54], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.6e54

   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. Step-by-step derivation

      1. associate-/l* 98.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}} \]

      2. fma-def 98.0%

         \[\leadsto \frac{x}{\frac{y}{e^{\color{blue}{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right)} - b}}} \]

      3. sub-neg 98.0%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \color{blue}{\left(t + \left(-1\right)\right)} \cdot \log a\right) - b}}} \]

      4. metadata-eval 98.0%

         \[\leadsto \frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + \color{blue}{-1}\right) \cdot \log a\right) - b}}} \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\mathsf{fma}\left(y, \log z, \left(t + -1\right) \cdot \log a\right) - b}}}} \]

   4. Taylor expanded in t around 0 83.0%

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(-1 \cdot \log a + y \cdot \log z\right) - b}}{y}} \]

   5. Taylor expanded in b around 0 64.1%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-1 \cdot \log a + y \cdot \log z}}}{y} \]
   6. Step-by-step derivation

      1. +-commutative 64.1%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y} \]

      2. mul-1-neg 64.1%

         \[\leadsto \frac{x \cdot e^{y \cdot \log z + \color{blue}{\left(-\log a\right)}}}{y} \]

      3. sub-neg 64.1%

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z - \log a}}}{y} \]

      4. exp-diff 64.1%

         \[\leadsto \frac{x \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{\log a}}}}{y} \]

      5. *-commutative 64.1%

         \[\leadsto \frac{x \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{\log a}}}{y} \]

      6. exp-to-pow 64.1%

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{z}^{y}}}{e^{\log a}}}{y} \]

      7. rem-exp-log 64.8%

         \[\leadsto \frac{x \cdot \frac{{z}^{y}}{\color{blue}{a}}}{y} \]

      8. *-commutative 64.8%

         \[\leadsto \frac{\color{blue}{\frac{{z}^{y}}{a} \cdot x}}{y} \]

   7. Simplified 64.8%

      \[\leadsto \frac{\color{blue}{\frac{{z}^{y}}{a} \cdot x}}{y} \]

   8. Taylor expanded in y around 0 33.3%

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

   if 1.6e54 < z

   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. Step-by-step derivation

      1. associate-*l/ 88.9%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

      2. *-commutative 88.9%

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      3. +-commutative 88.9%

         \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

      4. associate--l+ 88.9%

         \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

      5. exp-sum 75.8%

         \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

      6. *-commutative 75.8%

         \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      7. exp-to-pow 76.1%

         \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      8. sub-neg 76.1%

         \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      9. metadata-eval 76.1%

         \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

      10. exp-diff 67.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

      11. *-commutative 67.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

      12. exp-to-pow 67.9%

          \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   3. Simplified 67.9%

      \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

   4. Taylor expanded in t around 0 69.1%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 69.1%

         \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

      2. times-frac 69.6%

         \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   6. Simplified 69.6%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

   7. Taylor expanded in y around 0 59.7%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   8. Taylor expanded in b around 0 28.0%

      \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]
   9. Step-by-step derivation

      1. *-commutative 28.0%

         \[\leadsto \frac{x}{\color{blue}{y \cdot a}} \]

   10. Simplified 28.0%

       \[\leadsto \frac{x}{\color{blue}{y \cdot a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]

Alternative 27: 30.9% accurate, 63.0× 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 98.9%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
2. Step-by-step derivation

   1. associate-*l/ 92.0%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]

   2. *-commutative 92.0%

      \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

   3. +-commutative 92.0%

      \[\leadsto e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b} \cdot \frac{x}{y} \]

   4. associate--l+ 92.0%

      \[\leadsto e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z - b\right)}} \cdot \frac{x}{y} \]

   5. exp-sum 77.1%

      \[\leadsto \color{blue}{\left(e^{\left(t - 1\right) \cdot \log a} \cdot e^{y \cdot \log z - b}\right)} \cdot \frac{x}{y} \]

   6. *-commutative 77.1%

      \[\leadsto \left(e^{\color{blue}{\log a \cdot \left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

   7. exp-to-pow 77.6%

      \[\leadsto \left(\color{blue}{{a}^{\left(t - 1\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

   8. sub-neg 77.6%

      \[\leadsto \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

   9. metadata-eval 77.6%

      \[\leadsto \left({a}^{\left(t + \color{blue}{-1}\right)} \cdot e^{y \cdot \log z - b}\right) \cdot \frac{x}{y} \]

   10. exp-diff 69.8%

       \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{e^{y \cdot \log z}}{e^{b}}}\right) \cdot \frac{x}{y} \]

   11. *-commutative 69.8%

       \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{e^{\color{blue}{\log z \cdot y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

   12. exp-to-pow 69.8%

       \[\leadsto \left({a}^{\left(t + -1\right)} \cdot \frac{\color{blue}{{z}^{y}}}{e^{b}}\right) \cdot \frac{x}{y} \]

3. Simplified 69.8%

   \[\leadsto \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}\right) \cdot \frac{x}{y}} \]

4. Taylor expanded in t around 0 68.1%

   \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
5. Step-by-step derivation

   1. *-commutative 68.1%

      \[\leadsto \frac{\color{blue}{{z}^{y} \cdot x}}{a \cdot \left(y \cdot e^{b}\right)} \]

   2. times-frac 70.7%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

6. Simplified 70.7%

   \[\leadsto \color{blue}{\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}} \]

7. Taylor expanded in y around 0 59.8%

   \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

8. Taylor expanded in b around 0 28.2%

   \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]
9. Step-by-step derivation

   1. *-commutative 28.2%

      \[\leadsto \frac{x}{\color{blue}{y \cdot a}} \]

10. Simplified 28.2%

    \[\leadsto \frac{x}{\color{blue}{y \cdot a}} \]

11. Final simplification 28.2%

    \[\leadsto \frac{x}{y \cdot a} \]

Developer target: 71.9% 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 2023297 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))