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

Percentage Accurate: 98.3% → 98.3%

Time: 13.9s

Alternatives: 16

Speedup: 1.0×

• 47.3% 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: 98.3% 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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

3. Final simplification 98.7%

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

Alternative 2: 36.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y}\\ t_2 := x - b \cdot x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{\frac{a}{t\_2} \cdot y}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{x}{a \cdot y} \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_2}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* (exp (- (+ (* (log a) (- t 1.0)) (* (log z) y)) b)) x) y))
        (t_2 (- x (* b x))))
   (if (<= t_1 -2e+25)
     (/ 1.0 (* (/ a t_2) y))
     (if (<= t_1 0.0) (* (/ x (* a y)) (- b)) (/ (/ t_2 a) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp((((log(a) * (t - 1.0)) + (log(z) * y)) - b)) * x) / y;
	double t_2 = x - (b * x);
	double tmp;
	if (t_1 <= -2e+25) {
		tmp = 1.0 / ((a / t_2) * y);
	} else if (t_1 <= 0.0) {
		tmp = (x / (a * y)) * -b;
	} else {
		tmp = (t_2 / 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) :: t_2
    real(8) :: tmp
    t_1 = (exp((((log(a) * (t - 1.0d0)) + (log(z) * y)) - b)) * x) / y
    t_2 = x - (b * x)
    if (t_1 <= (-2d+25)) then
        tmp = 1.0d0 / ((a / t_2) * y)
    else if (t_1 <= 0.0d0) then
        tmp = (x / (a * y)) * -b
    else
        tmp = (t_2 / 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 = (Math.exp((((Math.log(a) * (t - 1.0)) + (Math.log(z) * y)) - b)) * x) / y;
	double t_2 = x - (b * x);
	double tmp;
	if (t_1 <= -2e+25) {
		tmp = 1.0 / ((a / t_2) * y);
	} else if (t_1 <= 0.0) {
		tmp = (x / (a * y)) * -b;
	} else {
		tmp = (t_2 / a) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.exp((((math.log(a) * (t - 1.0)) + (math.log(z) * y)) - b)) * x) / y
	t_2 = x - (b * x)
	tmp = 0
	if t_1 <= -2e+25:
		tmp = 1.0 / ((a / t_2) * y)
	elif t_1 <= 0.0:
		tmp = (x / (a * y)) * -b
	else:
		tmp = (t_2 / a) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(Float64(Float64(log(a) * Float64(t - 1.0)) + Float64(log(z) * y)) - b)) * x) / y)
	t_2 = Float64(x - Float64(b * x))
	tmp = 0.0
	if (t_1 <= -2e+25)
		tmp = Float64(1.0 / Float64(Float64(a / t_2) * y));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(x / Float64(a * y)) * Float64(-b));
	else
		tmp = Float64(Float64(t_2 / a) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (exp((((log(a) * (t - 1.0)) + (log(z) * y)) - b)) * x) / y;
	t_2 = x - (b * x);
	tmp = 0.0;
	if (t_1 <= -2e+25)
		tmp = 1.0 / ((a / t_2) * y);
	elseif (t_1 <= 0.0)
		tmp = (x / (a * y)) * -b;
	else
		tmp = (t_2 / a) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[N[(N[(N[(N[Log[a], $MachinePrecision] * N[(t - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(b * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+25], N[(1.0 / N[(N[(a / t$95$2), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision] * (-b)), $MachinePrecision], N[(N[(t$95$2 / a), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -2.00000000000000018e25

   1. Initial program 99.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 66.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 49.6%

       \[\leadsto \frac{\frac{x - b \cdot x}{a}}{y} \]
   12. Step-by-step derivation

   13. Applied rewrites 49.6%

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

   if -2.00000000000000018e25 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -0.0

   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. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 43.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 19.6%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 27.8%

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

   if -0.0 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 56.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 40.4%

       \[\leadsto \frac{\frac{x - b \cdot x}{a}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{\frac{a}{x - b \cdot x} \cdot y}\\ \mathbf{elif}\;\frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y} \leq 0:\\ \;\;\;\;\frac{x}{a \cdot y} \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - b \cdot x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 36.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y}\\ t_2 := \frac{\frac{x - b \cdot x}{a}}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+25}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{x}{a \cdot y} \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* (exp (- (+ (* (log a) (- t 1.0)) (* (log z) y)) b)) x) y))
        (t_2 (/ (/ (- x (* b x)) a) y)))
   (if (<= t_1 -2e+25) t_2 (if (<= t_1 0.0) (* (/ x (* a y)) (- b)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp((((log(a) * (t - 1.0)) + (log(z) * y)) - b)) * x) / y;
	double t_2 = ((x - (b * x)) / a) / y;
	double tmp;
	if (t_1 <= -2e+25) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (x / (a * y)) * -b;
	} 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 = (exp((((log(a) * (t - 1.0d0)) + (log(z) * y)) - b)) * x) / y
    t_2 = ((x - (b * x)) / a) / y
    if (t_1 <= (-2d+25)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = (x / (a * y)) * -b
    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.exp((((Math.log(a) * (t - 1.0)) + (Math.log(z) * y)) - b)) * x) / y;
	double t_2 = ((x - (b * x)) / a) / y;
	double tmp;
	if (t_1 <= -2e+25) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (x / (a * y)) * -b;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.exp((((math.log(a) * (t - 1.0)) + (math.log(z) * y)) - b)) * x) / y
	t_2 = ((x - (b * x)) / a) / y
	tmp = 0
	if t_1 <= -2e+25:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = (x / (a * y)) * -b
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(Float64(Float64(log(a) * Float64(t - 1.0)) + Float64(log(z) * y)) - b)) * x) / y)
	t_2 = Float64(Float64(Float64(x - Float64(b * x)) / a) / y)
	tmp = 0.0
	if (t_1 <= -2e+25)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(x / Float64(a * y)) * Float64(-b));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (exp((((log(a) * (t - 1.0)) + (log(z) * y)) - b)) * x) / y;
	t_2 = ((x - (b * x)) / a) / y;
	tmp = 0.0;
	if (t_1 <= -2e+25)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = (x / (a * y)) * -b;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[N[(N[(N[(N[Log[a], $MachinePrecision] * N[(t - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x - N[(b * x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+25], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision] * (-b)), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -2.00000000000000018e25 or -0.0 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   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. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 61.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 45.2%

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

   if -2.00000000000000018e25 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -0.0

   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. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 43.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 19.6%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 27.8%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{x - b \cdot x}{a}}{y}\\ \mathbf{elif}\;\frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y} \leq 0:\\ \;\;\;\;\frac{x}{a \cdot y} \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - b \cdot x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{\log z \cdot y} \cdot x}{y}\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+104}:\\ \;\;\;\;\frac{e^{-\mathsf{fma}\left(\log a, 1 - t, b\right)}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* (exp (* (log z) y)) x) y)))
   (if (<= y -1.5e+155)
     t_1
     (if (<= y 5.8e+104)
       (* (/ (exp (- (fma (log a) (- 1.0 t) b))) y) x)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp((log(z) * y)) * x) / y;
	double tmp;
	if (y <= -1.5e+155) {
		tmp = t_1;
	} else if (y <= 5.8e+104) {
		tmp = (exp(-fma(log(a), (1.0 - t), b)) / y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(log(z) * y)) * x) / y)
	tmp = 0.0
	if (y <= -1.5e+155)
		tmp = t_1;
	elseif (y <= 5.8e+104)
		tmp = Float64(Float64(exp(Float64(-fma(log(a), Float64(1.0 - t), b))) / y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -1.5e+155], t$95$1, If[LessEqual[y, 5.8e+104], N[(N[(N[Exp[(-N[(N[Log[a], $MachinePrecision] * N[(1.0 - t), $MachinePrecision] + b), $MachinePrecision])], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5000000000000001e155 or 5.7999999999999997e104 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if -1.5000000000000001e155 < y < 5.7999999999999997e104

   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. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-lft-neg-out N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. lower-neg.f64 N/A

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

      9. log-rec N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 94.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 94.5%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{e^{\log z \cdot y} \cdot x}{y}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+104}:\\ \;\;\;\;\frac{e^{-\mathsf{fma}\left(\log a, 1 - t, b\right)}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log z \cdot y} \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{\log z \cdot y} \cdot x}{y}\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{e^{b} \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* (exp (* (log z) y)) x) y)))
   (if (<= y -4.6e+137)
     t_1
     (if (<= y 5.6e+104) (* (/ (pow a (- t 1.0)) (* (exp b) y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp((log(z) * y)) * x) / y;
	double tmp;
	if (y <= -4.6e+137) {
		tmp = t_1;
	} else if (y <= 5.6e+104) {
		tmp = (pow(a, (t - 1.0)) / (exp(b) * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (exp((log(z) * y)) * x) / y
    if (y <= (-4.6d+137)) then
        tmp = t_1
    else if (y <= 5.6d+104) then
        tmp = ((a ** (t - 1.0d0)) / (exp(b) * y)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (Math.exp((Math.log(z) * y)) * x) / y;
	double tmp;
	if (y <= -4.6e+137) {
		tmp = t_1;
	} else if (y <= 5.6e+104) {
		tmp = (Math.pow(a, (t - 1.0)) / (Math.exp(b) * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.exp((math.log(z) * y)) * x) / y
	tmp = 0
	if y <= -4.6e+137:
		tmp = t_1
	elif y <= 5.6e+104:
		tmp = (math.pow(a, (t - 1.0)) / (math.exp(b) * y)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(log(z) * y)) * x) / y)
	tmp = 0.0
	if (y <= -4.6e+137)
		tmp = t_1;
	elseif (y <= 5.6e+104)
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) / Float64(exp(b) * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (exp((log(z) * y)) * x) / y;
	tmp = 0.0;
	if (y <= -4.6e+137)
		tmp = t_1;
	elseif (y <= 5.6e+104)
		tmp = ((a ^ (t - 1.0)) / (exp(b) * y)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -4.6e+137], t$95$1, If[LessEqual[y, 5.6e+104], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.59999999999999999e137 or 5.6e104 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if -4.59999999999999999e137 < y < 5.6e104

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-lft-neg-out N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. lower-neg.f64 N/A

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

      9. log-rec N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 94.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 95.0%

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

   8. Taylor expanded in y around 0

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

      1. exp-diff N/A

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

      2. associate-/l/ N/A

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

      3. lower-/.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 88.9

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

   10. Applied rewrites 88.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+137}:\\ \;\;\;\;\frac{e^{\log z \cdot y} \cdot x}{y}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{e^{b} \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log z \cdot y} \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{\log z \cdot y} \cdot x}{y}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 280000:\\ \;\;\;\;\frac{e^{\log a \cdot t - b} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* (exp (* (log z) y)) x) y)))
   (if (<= y -6.5e+155)
     t_1
     (if (<= y 280000.0) (/ (* (exp (- (* (log a) t) b)) x) y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp((log(z) * y)) * x) / y;
	double tmp;
	if (y <= -6.5e+155) {
		tmp = t_1;
	} else if (y <= 280000.0) {
		tmp = (exp(((log(a) * t) - b)) * x) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (exp((log(z) * y)) * x) / y
    if (y <= (-6.5d+155)) then
        tmp = t_1
    else if (y <= 280000.0d0) then
        tmp = (exp(((log(a) * t) - b)) * x) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (Math.exp((Math.log(z) * y)) * x) / y;
	double tmp;
	if (y <= -6.5e+155) {
		tmp = t_1;
	} else if (y <= 280000.0) {
		tmp = (Math.exp(((Math.log(a) * t) - b)) * x) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.exp((math.log(z) * y)) * x) / y
	tmp = 0
	if y <= -6.5e+155:
		tmp = t_1
	elif y <= 280000.0:
		tmp = (math.exp(((math.log(a) * t) - b)) * x) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(log(z) * y)) * x) / y)
	tmp = 0.0
	if (y <= -6.5e+155)
		tmp = t_1;
	elseif (y <= 280000.0)
		tmp = Float64(Float64(exp(Float64(Float64(log(a) * t) - b)) * x) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (exp((log(z) * y)) * x) / y;
	tmp = 0.0;
	if (y <= -6.5e+155)
		tmp = t_1;
	elseif (y <= 280000.0)
		tmp = (exp(((log(a) * t) - b)) * x) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -6.5e+155], t$95$1, If[LessEqual[y, 280000.0], N[(N[(N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.50000000000000046e155 or 2.8e5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if -6.50000000000000046e155 < y < 2.8e5

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lower-log.f64 N/A

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

      4. rem-exp-log 82.5

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

   5. Applied rewrites 82.5%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{e^{\log z \cdot y} \cdot x}{y}\\ \mathbf{elif}\;y \leq 280000:\\ \;\;\;\;\frac{e^{\log a \cdot t - b} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log z \cdot y} \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - b \cdot x\\ \mathbf{if}\;\log a \leq -245:\\ \;\;\;\;\frac{\frac{t\_1}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot y}{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- x (* b x))))
   (if (<= (log a) -245.0) (/ (/ t_1 a) y) (/ 1.0 (/ (* a y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x - (b * x);
	double tmp;
	if (log(a) <= -245.0) {
		tmp = (t_1 / a) / y;
	} else {
		tmp = 1.0 / ((a * y) / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x - (b * x)
    if (log(a) <= (-245.0d0)) then
        tmp = (t_1 / a) / y
    else
        tmp = 1.0d0 / ((a * y) / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x - (b * x);
	double tmp;
	if (Math.log(a) <= -245.0) {
		tmp = (t_1 / a) / y;
	} else {
		tmp = 1.0 / ((a * y) / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x - (b * x)
	tmp = 0
	if math.log(a) <= -245.0:
		tmp = (t_1 / a) / y
	else:
		tmp = 1.0 / ((a * y) / t_1)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x - Float64(b * x))
	tmp = 0.0
	if (log(a) <= -245.0)
		tmp = Float64(Float64(t_1 / a) / y);
	else
		tmp = Float64(1.0 / Float64(Float64(a * y) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x - (b * x);
	tmp = 0.0;
	if (log(a) <= -245.0)
		tmp = (t_1 / a) / y;
	else
		tmp = 1.0 / ((a * y) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x - N[(b * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Log[a], $MachinePrecision], -245.0], N[(N[(t$95$1 / a), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(N[(a * y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 a) < -245

   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. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 59.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 40.2%

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

   if -245 < (log.f64 a)

   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. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 49.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 29.5%

       \[\leadsto \frac{\frac{x - b \cdot x}{a}}{y} \]
   12. Step-by-step derivation

   13. Applied rewrites 34.4%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \leq -245:\\ \;\;\;\;\frac{\frac{x - b \cdot x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot y}{x - b \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{-b}}{y} \cdot x\\ \mathbf{if}\;b \leq -2.55 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 420:\\ \;\;\;\;\left(x - b \cdot x\right) \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ (exp (- b)) y) x)))
   (if (<= b -2.55e+20)
     t_1
     (if (<= b 420.0) (* (- x (* b x)) (/ (pow a (- t 1.0)) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp(-b) / y) * x;
	double tmp;
	if (b <= -2.55e+20) {
		tmp = t_1;
	} else if (b <= 420.0) {
		tmp = (x - (b * x)) * (pow(a, (t - 1.0)) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (exp(-b) / y) * x
    if (b <= (-2.55d+20)) then
        tmp = t_1
    else if (b <= 420.0d0) then
        tmp = (x - (b * x)) * ((a ** (t - 1.0d0)) / y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (Math.exp(-b) / y) * x;
	double tmp;
	if (b <= -2.55e+20) {
		tmp = t_1;
	} else if (b <= 420.0) {
		tmp = (x - (b * x)) * (Math.pow(a, (t - 1.0)) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.exp(-b) / y) * x
	tmp = 0
	if b <= -2.55e+20:
		tmp = t_1
	elif b <= 420.0:
		tmp = (x - (b * x)) * (math.pow(a, (t - 1.0)) / y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(-b)) / y) * x)
	tmp = 0.0
	if (b <= -2.55e+20)
		tmp = t_1;
	elseif (b <= 420.0)
		tmp = Float64(Float64(x - Float64(b * x)) * Float64((a ^ Float64(t - 1.0)) / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (exp(-b) / y) * x;
	tmp = 0.0;
	if (b <= -2.55e+20)
		tmp = t_1;
	elseif (b <= 420.0)
		tmp = (x - (b * x)) * ((a ^ (t - 1.0)) / y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -2.55e+20], t$95$1, If[LessEqual[b, 420.0], N[(N[(x - N[(b * x), $MachinePrecision]), $MachinePrecision] * N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.55e20 or 420 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 86.5

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

   5. Applied rewrites 86.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.5

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

   7. Applied rewrites 86.5%

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

   if -2.55e20 < b < 420

   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. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 75.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{+20}:\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{elif}\;b \leq 420:\\ \;\;\;\;\left(x - b \cdot x\right) \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{-b}}{y} \cdot x\\ \mathbf{if}\;b \leq -6.6 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 65000000000:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ (exp (- b)) y) x)))
   (if (<= b -6.6e+22)
     t_1
     (if (<= b 65000000000.0) (* (/ (pow a (- t 1.0)) y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp(-b) / y) * x;
	double tmp;
	if (b <= -6.6e+22) {
		tmp = t_1;
	} else if (b <= 65000000000.0) {
		tmp = (pow(a, (t - 1.0)) / y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (exp(-b) / y) * x
    if (b <= (-6.6d+22)) then
        tmp = t_1
    else if (b <= 65000000000.0d0) then
        tmp = ((a ** (t - 1.0d0)) / y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (Math.exp(-b) / y) * x;
	double tmp;
	if (b <= -6.6e+22) {
		tmp = t_1;
	} else if (b <= 65000000000.0) {
		tmp = (Math.pow(a, (t - 1.0)) / y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.exp(-b) / y) * x
	tmp = 0
	if b <= -6.6e+22:
		tmp = t_1
	elif b <= 65000000000.0:
		tmp = (math.pow(a, (t - 1.0)) / y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(-b)) / y) * x)
	tmp = 0.0
	if (b <= -6.6e+22)
		tmp = t_1;
	elseif (b <= 65000000000.0)
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) / y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (exp(-b) / y) * x;
	tmp = 0.0;
	if (b <= -6.6e+22)
		tmp = t_1;
	elseif (b <= 65000000000.0)
		tmp = ((a ^ (t - 1.0)) / y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -6.6e+22], t$95$1, If[LessEqual[b, 65000000000.0], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.5999999999999996e22 or 6.5e10 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.0

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

   5. Applied rewrites 87.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.0

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

   7. Applied rewrites 87.0%

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

   if -6.5999999999999996e22 < b < 6.5e10

   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. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-lft-neg-out N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. lower-neg.f64 N/A

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

      9. log-rec N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 73.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 75.2%

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

   8. Taylor expanded in y around 0

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

      1. exp-diff N/A

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

      2. associate-/l/ N/A

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

      3. lower-/.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 74.5

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

   10. Applied rewrites 74.5%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 75.6%

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

Alternative 10: 57.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{-b}}{y} \cdot x\\ \mathbf{if}\;b \leq -6800000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{1}{\frac{a \cdot y}{x - b \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ (exp (- b)) y) x)))
   (if (<= b -6800000000.0)
     t_1
     (if (<= b 1.05e-85) (/ 1.0 (/ (* a y) (- x (* b x)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp(-b) / y) * x;
	double tmp;
	if (b <= -6800000000.0) {
		tmp = t_1;
	} else if (b <= 1.05e-85) {
		tmp = 1.0 / ((a * y) / (x - (b * x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (exp(-b) / y) * x
    if (b <= (-6800000000.0d0)) then
        tmp = t_1
    else if (b <= 1.05d-85) then
        tmp = 1.0d0 / ((a * y) / (x - (b * x)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (Math.exp(-b) / y) * x;
	double tmp;
	if (b <= -6800000000.0) {
		tmp = t_1;
	} else if (b <= 1.05e-85) {
		tmp = 1.0 / ((a * y) / (x - (b * x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.exp(-b) / y) * x
	tmp = 0
	if b <= -6800000000.0:
		tmp = t_1
	elif b <= 1.05e-85:
		tmp = 1.0 / ((a * y) / (x - (b * x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(-b)) / y) * x)
	tmp = 0.0
	if (b <= -6800000000.0)
		tmp = t_1;
	elseif (b <= 1.05e-85)
		tmp = Float64(1.0 / Float64(Float64(a * y) / Float64(x - Float64(b * x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (exp(-b) / y) * x;
	tmp = 0.0;
	if (b <= -6800000000.0)
		tmp = t_1;
	elseif (b <= 1.05e-85)
		tmp = 1.0 / ((a * y) / (x - (b * x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -6800000000.0], t$95$1, If[LessEqual[b, 1.05e-85], N[(1.0 / N[(N[(a * y), $MachinePrecision] / N[(x - N[(b * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.8e9 or 1.05e-85 < b

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 80.5

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

   5. Applied rewrites 80.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.5

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

   7. Applied rewrites 80.5%

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

   if -6.8e9 < b < 1.05e-85

   1. Initial program 97.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 68.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 42.3%

       \[\leadsto \frac{\frac{x - b \cdot x}{a}}{y} \]
   12. Step-by-step derivation

   13. Applied rewrites 45.5%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6800000000:\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{1}{\frac{a \cdot y}{x - b \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} \cdot e^{-b}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{1}{\frac{a \cdot y}{x - b \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ x y) (exp (- b)))))
   (if (<= b -2e+20)
     t_1
     (if (<= b 1.05e-85) (/ 1.0 (/ (* a y) (- x (* b x)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x / y) * exp(-b);
	double tmp;
	if (b <= -2e+20) {
		tmp = t_1;
	} else if (b <= 1.05e-85) {
		tmp = 1.0 / ((a * y) / (x - (b * x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / y) * exp(-b)
    if (b <= (-2d+20)) then
        tmp = t_1
    else if (b <= 1.05d-85) then
        tmp = 1.0d0 / ((a * y) / (x - (b * x)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x / y) * Math.exp(-b);
	double tmp;
	if (b <= -2e+20) {
		tmp = t_1;
	} else if (b <= 1.05e-85) {
		tmp = 1.0 / ((a * y) / (x - (b * x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x / y) * math.exp(-b)
	tmp = 0
	if b <= -2e+20:
		tmp = t_1
	elif b <= 1.05e-85:
		tmp = 1.0 / ((a * y) / (x - (b * x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x / y) * exp(Float64(-b)))
	tmp = 0.0
	if (b <= -2e+20)
		tmp = t_1;
	elseif (b <= 1.05e-85)
		tmp = Float64(1.0 / Float64(Float64(a * y) / Float64(x - Float64(b * x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x / y) * exp(-b);
	tmp = 0.0;
	if (b <= -2e+20)
		tmp = t_1;
	elseif (b <= 1.05e-85)
		tmp = 1.0 / ((a * y) / (x - (b * x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] * N[Exp[(-b)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+20], t$95$1, If[LessEqual[b, 1.05e-85], N[(1.0 / N[(N[(a * y), $MachinePrecision] / N[(x - N[(b * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -2e20 or 1.05e-85 < b

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.6

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

   5. Applied rewrites 81.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.3

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

   7. Applied rewrites 73.3%

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

   if -2e20 < b < 1.05e-85

   1. Initial program 97.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 41.4%

       \[\leadsto \frac{\frac{x - b \cdot x}{a}}{y} \]
   12. Step-by-step derivation

   13. Applied rewrites 44.4%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{y} \cdot e^{-b}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{1}{\frac{a \cdot y}{x - b \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot e^{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1 - b}{a} \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{x}{a} \cdot y - \left(\frac{x}{a} \cdot b\right) \cdot y}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - b \cdot x}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 5.4e-14)
   (* (/ (- 1.0 b) a) (/ x y))
   (if (<= x 1.45e+116)
     (/ (- (* (/ x a) y) (* (* (/ x a) b) y)) (* y y))
     (/ (/ (- x (* b x)) a) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 5.4e-14) {
		tmp = ((1.0 - b) / a) * (x / y);
	} else if (x <= 1.45e+116) {
		tmp = (((x / a) * y) - (((x / a) * b) * y)) / (y * y);
	} else {
		tmp = ((x - (b * x)) / 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 (x <= 5.4d-14) then
        tmp = ((1.0d0 - b) / a) * (x / y)
    else if (x <= 1.45d+116) then
        tmp = (((x / a) * y) - (((x / a) * b) * y)) / (y * y)
    else
        tmp = ((x - (b * x)) / 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 (x <= 5.4e-14) {
		tmp = ((1.0 - b) / a) * (x / y);
	} else if (x <= 1.45e+116) {
		tmp = (((x / a) * y) - (((x / a) * b) * y)) / (y * y);
	} else {
		tmp = ((x - (b * x)) / a) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 5.4e-14:
		tmp = ((1.0 - b) / a) * (x / y)
	elif x <= 1.45e+116:
		tmp = (((x / a) * y) - (((x / a) * b) * y)) / (y * y)
	else:
		tmp = ((x - (b * x)) / a) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 5.4e-14)
		tmp = Float64(Float64(Float64(1.0 - b) / a) * Float64(x / y));
	elseif (x <= 1.45e+116)
		tmp = Float64(Float64(Float64(Float64(x / a) * y) - Float64(Float64(Float64(x / a) * b) * y)) / Float64(y * y));
	else
		tmp = Float64(Float64(Float64(x - Float64(b * x)) / a) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 5.4e-14)
		tmp = ((1.0 - b) / a) * (x / y);
	elseif (x <= 1.45e+116)
		tmp = (((x / a) * y) - (((x / a) * b) * y)) / (y * y);
	else
		tmp = ((x - (b * x)) / a) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 5.4e-14], N[(N[(N[(1.0 - b), $MachinePrecision] / a), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e+116], N[(N[(N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(x / a), $MachinePrecision] * b), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(b * x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.3999999999999997e-14

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 52.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 33.0%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 34.2%

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

   if 5.3999999999999997e-14 < x < 1.4500000000000001e116

   1. Initial program 99.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 60.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 44.5%

       \[\leadsto \frac{\frac{x - b \cdot x}{a}}{y} \]
   12. Step-by-step derivation

   13. Applied rewrites 56.9%

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

   if 1.4500000000000001e116 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 52.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 27.9%

       \[\leadsto \frac{\frac{x - b \cdot x}{a}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1 - b}{a} \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{x}{a} \cdot y - \left(\frac{x}{a} \cdot b\right) \cdot y}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - b \cdot x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 33.5% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{1 - b}{a} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - b \cdot x}{a \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 1.6e-14) (* (/ (- 1.0 b) a) (/ x y)) (/ (- x (* b x)) (* a y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 1.6e-14) {
		tmp = ((1.0 - b) / a) * (x / y);
	} else {
		tmp = (x - (b * x)) / (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 (x <= 1.6d-14) then
        tmp = ((1.0d0 - b) / a) * (x / y)
    else
        tmp = (x - (b * x)) / (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 (x <= 1.6e-14) {
		tmp = ((1.0 - b) / a) * (x / y);
	} else {
		tmp = (x - (b * x)) / (a * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 1.6e-14:
		tmp = ((1.0 - b) / a) * (x / y)
	else:
		tmp = (x - (b * x)) / (a * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 1.6e-14)
		tmp = Float64(Float64(Float64(1.0 - b) / a) * Float64(x / y));
	else
		tmp = Float64(Float64(x - Float64(b * x)) / Float64(a * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 1.6e-14)
		tmp = ((1.0 - b) / a) * (x / y);
	else
		tmp = (x - (b * x)) / (a * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 1.6e-14], N[(N[(N[(1.0 - b), $MachinePrecision] / a), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(b * x), $MachinePrecision]), $MachinePrecision] / N[(a * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001e-14

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 52.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 33.0%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 34.3%

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

   if 1.6000000000000001e-14 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 55.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 34.8%

       \[\leadsto \frac{\frac{x - b \cdot x}{a}}{y} \]
   12. Step-by-step derivation

   13. Applied rewrites 35.3%

       \[\leadsto \frac{x - x \cdot b}{a \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{1 - b}{a} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - b \cdot x}{a \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 34.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ \mathbf{if}\;b \leq -1.18 \cdot 10^{+21}:\\ \;\;\;\;t\_1 \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* a y)))) (if (<= b -1.18e+21) (* t_1 (- b)) t_1)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a * y);
	double tmp;
	if (b <= -1.18e+21) {
		tmp = t_1 * -b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (a * y)
    if (b <= (-1.18d+21)) then
        tmp = t_1 * -b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a * y);
	double tmp;
	if (b <= -1.18e+21) {
		tmp = t_1 * -b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (a * y)
	tmp = 0
	if b <= -1.18e+21:
		tmp = t_1 * -b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a * y))
	tmp = 0.0
	if (b <= -1.18e+21)
		tmp = Float64(t_1 * Float64(-b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (a * y);
	tmp = 0.0;
	if (b <= -1.18e+21)
		tmp = t_1 * -b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.18e+21], N[(t$95$1 * (-b)), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if b < -1.18e21

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 54.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 57.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 42.3%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 33.1%

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

   if -1.18e21 < b

   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. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 51.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 30.6%

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

   12. Taylor expanded in b around 0

       \[\leadsto \frac{x}{a \cdot y} \]
   13. Step-by-step derivation

   14. Applied rewrites 33.7%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{a \cdot y} \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 32.8% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \frac{x - b \cdot x}{a \cdot y} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x - (b * x)) / (a * 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 - (b * x)) / (a * y)
end function

Java

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

Python

def code(x, y, z, t, a, b):
	return (x - (b * x)) / (a * y)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. associate-/l* N/A

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

   5. associate-*r* N/A

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

   6. associate-/l* N/A

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

   7. distribute-rgt-out-- N/A

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

   8. lower-*.f64 N/A

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

5. Applied rewrites 63.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 53.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 33.4%

    \[\leadsto \frac{\frac{x - b \cdot x}{a}}{y} \]
12. Step-by-step derivation

13. Applied rewrites 33.6%

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

14. Final simplification 33.6%

    \[\leadsto \frac{x - b \cdot x}{a \cdot y} \]
15. Add Preprocessing

Alternative 16: 31.5% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{a \cdot y} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / (a * 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 / (a * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. associate-/l* N/A

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

   5. associate-*r* N/A

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

   6. associate-/l* N/A

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

   7. distribute-rgt-out-- N/A

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

   8. lower-*.f64 N/A

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

5. Applied rewrites 63.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 53.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 33.4%

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

12. Taylor expanded in b around 0

    \[\leadsto \frac{x}{a \cdot y} \]
13. Step-by-step derivation

14. Applied rewrites 30.0%

    \[\leadsto \frac{x}{a \cdot y} \]
15. Add Preprocessing

Developer Target 1: 72.4% 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 2024243 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -8845848504127471/10000000000000000) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 8520312288374073/10000000000) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))