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

Percentage Accurate: 98.3% → 98.3%

Time: 19.1s

Alternatives: 17

Speedup: 1.0×

• 20.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.8%

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

2. Final simplification 97.8%

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

Alternative 2: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t + -1 \leq -1 \cdot 10^{+122} \lor \neg \left(t + -1 \leq 5 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (+ t -1.0) -1e+122) (not (<= (+ t -1.0) 5e+69)))
   (/ (* x (pow a (+ t -1.0))) y)
   (* x (/ (pow z y) (* y (* a (exp b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((t + -1.0) <= -1e+122) || !((t + -1.0) <= 5e+69)) {
		tmp = (x * pow(a, (t + -1.0))) / y;
	} else {
		tmp = x * (pow(z, y) / (y * (a * exp(b))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((t + (-1.0d0)) <= (-1d+122)) .or. (.not. ((t + (-1.0d0)) <= 5d+69))) then
        tmp = (x * (a ** (t + (-1.0d0)))) / y
    else
        tmp = x * ((z ** y) / (y * (a * exp(b))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((t + -1.0) <= -1e+122) || !((t + -1.0) <= 5e+69)) {
		tmp = (x * Math.pow(a, (t + -1.0))) / y;
	} else {
		tmp = x * (Math.pow(z, y) / (y * (a * Math.exp(b))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((t + -1.0) <= -1e+122) or not ((t + -1.0) <= 5e+69):
		tmp = (x * math.pow(a, (t + -1.0))) / y
	else:
		tmp = x * (math.pow(z, y) / (y * (a * math.exp(b))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(t + -1.0) <= -1e+122) || !(Float64(t + -1.0) <= 5e+69))
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	else
		tmp = Float64(x * Float64((z ^ y) / Float64(y * Float64(a * exp(b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((t + -1.0) <= -1e+122) || ~(((t + -1.0) <= 5e+69)))
		tmp = (x * (a ^ (t + -1.0))) / y;
	else
		tmp = x * ((z ^ y) / (y * (a * exp(b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(t + -1.0), $MachinePrecision], -1e+122], N[Not[LessEqual[N[(t + -1.0), $MachinePrecision], 5e+69]], $MachinePrecision]], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(N[Power[z, y], $MachinePrecision] / N[(y * N[(a * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 t 1) < -1.00000000000000001e122 or 5.00000000000000036e69 < (-.f64 t 1)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 92.3%

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

   3. Taylor expanded in b around 0 90.0%

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

   if -1.00000000000000001e122 < (-.f64 t 1) < 5.00000000000000036e69

   1. Initial program 96.6%

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

      1. associate-*r/ 96.5%

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

      2. sub-neg 96.5%

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

      3. exp-sum 81.5%

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

      4. associate-/l* 81.5%

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

      5. associate-/r/ 80.3%

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

      6. exp-neg 80.3%

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

      7. associate-*r/ 80.3%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in t around 0 81.7%

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

      1. *-commutative 81.7%

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

      2. associate-*l* 81.7%

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

      3. *-commutative 81.7%

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

   6. Simplified 81.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t + -1 \leq -1 \cdot 10^{+122} \lor \neg \left(t + -1 \leq 5 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \]

Alternative 3: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t + -1 \leq -1 \cdot 10^{+122} \lor \neg \left(t + -1 \leq 5 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (+ t -1.0) -1e+122) (not (<= (+ t -1.0) 5e+69)))
   (/ (* x (pow a (+ t -1.0))) y)
   (/ (* x (pow z y)) (* y (* a (exp b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((t + -1.0) <= -1e+122) || !((t + -1.0) <= 5e+69)) {
		tmp = (x * pow(a, (t + -1.0))) / y;
	} else {
		tmp = (x * pow(z, y)) / (y * (a * exp(b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((t + (-1.0d0)) <= (-1d+122)) .or. (.not. ((t + (-1.0d0)) <= 5d+69))) then
        tmp = (x * (a ** (t + (-1.0d0)))) / y
    else
        tmp = (x * (z ** y)) / (y * (a * exp(b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((t + -1.0) <= -1e+122) || !((t + -1.0) <= 5e+69)) {
		tmp = (x * Math.pow(a, (t + -1.0))) / y;
	} else {
		tmp = (x * Math.pow(z, y)) / (y * (a * Math.exp(b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((t + -1.0) <= -1e+122) or not ((t + -1.0) <= 5e+69):
		tmp = (x * math.pow(a, (t + -1.0))) / y
	else:
		tmp = (x * math.pow(z, y)) / (y * (a * math.exp(b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(t + -1.0) <= -1e+122) || !(Float64(t + -1.0) <= 5e+69))
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	else
		tmp = Float64(Float64(x * (z ^ y)) / Float64(y * Float64(a * exp(b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((t + -1.0) <= -1e+122) || ~(((t + -1.0) <= 5e+69)))
		tmp = (x * (a ^ (t + -1.0))) / y;
	else
		tmp = (x * (z ^ y)) / (y * (a * exp(b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(t + -1.0), $MachinePrecision], -1e+122], N[Not[LessEqual[N[(t + -1.0), $MachinePrecision], 5e+69]], $MachinePrecision]], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / N[(y * N[(a * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 t 1) < -1.00000000000000001e122 or 5.00000000000000036e69 < (-.f64 t 1)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 92.3%

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

   3. Taylor expanded in b around 0 90.0%

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

   if -1.00000000000000001e122 < (-.f64 t 1) < 5.00000000000000036e69

   1. Initial program 96.6%

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

      1. associate-*r/ 96.5%

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

      2. sub-neg 96.5%

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

      3. exp-sum 81.5%

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

      4. associate-/l* 81.5%

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

      5. associate-/r/ 80.3%

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

      6. exp-neg 80.3%

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

      7. associate-*r/ 80.3%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in t around 0 82.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t + -1 \leq -1 \cdot 10^{+122} \lor \neg \left(t + -1 \leq 5 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \]

Alternative 4: 88.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+41} \lor \neg \left(y \leq 6.5 \cdot 10^{+64}\right):\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(t + -1\right) \cdot \log a - b}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -4.5e+41) (not (<= y 6.5e+64)))
   (* x (/ (/ (pow z y) a) y))
   (/ (* x (exp (- (* (+ t -1.0) (log a)) b))) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.5e+41) || !(y <= 6.5e+64)) {
		tmp = x * ((pow(z, y) / a) / y);
	} else {
		tmp = (x * exp((((t + -1.0) * log(a)) - b))) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-4.5d+41)) .or. (.not. (y <= 6.5d+64))) then
        tmp = x * (((z ** y) / a) / y)
    else
        tmp = (x * exp((((t + (-1.0d0)) * log(a)) - b))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.5e+41) || !(y <= 6.5e+64)) {
		tmp = x * ((Math.pow(z, y) / a) / y);
	} else {
		tmp = (x * Math.exp((((t + -1.0) * Math.log(a)) - b))) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -4.5e+41) or not (y <= 6.5e+64):
		tmp = x * ((math.pow(z, y) / a) / y)
	else:
		tmp = (x * math.exp((((t + -1.0) * math.log(a)) - b))) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -4.5e+41) || !(y <= 6.5e+64))
		tmp = Float64(x * Float64(Float64((z ^ y) / a) / y));
	else
		tmp = Float64(Float64(x * exp(Float64(Float64(Float64(t + -1.0) * log(a)) - b))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -4.5e+41) || ~((y <= 6.5e+64)))
		tmp = x * (((z ^ y) / a) / y);
	else
		tmp = (x * exp((((t + -1.0) * log(a)) - b))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -4.5e+41], N[Not[LessEqual[y, 6.5e+64]], $MachinePrecision]], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Exp[N[(N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5000000000000001e41 or 6.50000000000000007e64 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-sum 79.8%

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

      4. associate-/l* 79.8%

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

      5. associate-/r/ 79.8%

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

      6. exp-neg 79.8%

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

      7. associate-*r/ 79.8%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in t around 0 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-*l* 68.0%

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

      3. *-commutative 68.0%

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

   6. Simplified 68.0%

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

   7. Taylor expanded in b around 0 70.9%

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

      1. *-un-lft-identity 70.9%

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

      2. times-frac 84.6%

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

   9. Applied egg-rr 84.6%

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

       1. associate-*l/ 84.6%

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

       2. *-un-lft-identity 84.6%

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

   11. Applied egg-rr 84.6%

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

   if -4.5000000000000001e41 < y < 6.50000000000000007e64

   1. Initial program 96.1%

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

   2. Taylor expanded in y around 0 94.1%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+41} \lor \neg \left(y \leq 6.5 \cdot 10^{+64}\right):\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(t + -1\right) \cdot \log a - b}}{y}\\ \end{array} \]

Alternative 5: 72.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t + -1 \leq -2 \cdot 10^{+150} \lor \neg \left(t + -1 \leq 5 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (+ t -1.0) -2e+150) (not (<= (+ t -1.0) 5e+69)))
   (/ (* x (pow a (+ t -1.0))) y)
   (* x (/ (/ (pow z y) a) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((t + -1.0) <= -2e+150) || !((t + -1.0) <= 5e+69)) {
		tmp = (x * pow(a, (t + -1.0))) / y;
	} else {
		tmp = x * ((pow(z, y) / a) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((t + (-1.0d0)) <= (-2d+150)) .or. (.not. ((t + (-1.0d0)) <= 5d+69))) then
        tmp = (x * (a ** (t + (-1.0d0)))) / y
    else
        tmp = x * (((z ** y) / a) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((t + -1.0) <= -2e+150) || !((t + -1.0) <= 5e+69)) {
		tmp = (x * Math.pow(a, (t + -1.0))) / y;
	} else {
		tmp = x * ((Math.pow(z, y) / a) / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((t + -1.0) <= -2e+150) or not ((t + -1.0) <= 5e+69):
		tmp = (x * math.pow(a, (t + -1.0))) / y
	else:
		tmp = x * ((math.pow(z, y) / a) / y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(t + -1.0) <= -2e+150) || !(Float64(t + -1.0) <= 5e+69))
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	else
		tmp = Float64(x * Float64(Float64((z ^ y) / a) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((t + -1.0) <= -2e+150) || ~(((t + -1.0) <= 5e+69)))
		tmp = (x * (a ^ (t + -1.0))) / y;
	else
		tmp = x * (((z ^ y) / a) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(t + -1.0), $MachinePrecision], -2e+150], N[Not[LessEqual[N[(t + -1.0), $MachinePrecision], 5e+69]], $MachinePrecision]], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 t 1) < -1.99999999999999996e150 or 5.00000000000000036e69 < (-.f64 t 1)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 96.4%

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

   3. Taylor expanded in b around 0 94.1%

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

   if -1.99999999999999996e150 < (-.f64 t 1) < 5.00000000000000036e69

   1. Initial program 96.7%

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

      1. associate-*r/ 96.6%

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

      2. sub-neg 96.6%

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

      3. exp-sum 81.6%

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

      4. associate-/l* 81.6%

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

      5. associate-/r/ 80.4%

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

      6. exp-neg 80.4%

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

      7. associate-*r/ 80.4%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in t around 0 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-*l* 80.0%

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

      3. *-commutative 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in b around 0 67.1%

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

      1. *-un-lft-identity 67.1%

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

      2. times-frac 73.5%

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

   9. Applied egg-rr 73.5%

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

       1. associate-*l/ 73.5%

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

       2. *-un-lft-identity 73.5%

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

   11. Applied egg-rr 73.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t + -1 \leq -2 \cdot 10^{+150} \lor \neg \left(t + -1 \leq 5 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y}\\ \end{array} \]

Alternative 6: 74.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \lor \neg \left(y \leq 1.5 \cdot 10^{+27}\right):\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.1) (not (<= y 1.5e+27)))
   (* x (/ (/ (pow z y) a) y))
   (/ x (* y (* a (exp b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.1) || !(y <= 1.5e+27)) {
		tmp = x * ((pow(z, y) / a) / y);
	} else {
		tmp = x / (y * (a * exp(b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-2.1d0)) .or. (.not. (y <= 1.5d+27))) then
        tmp = x * (((z ** y) / a) / y)
    else
        tmp = x / (y * (a * exp(b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.1) || !(y <= 1.5e+27)) {
		tmp = x * ((Math.pow(z, y) / a) / y);
	} else {
		tmp = x / (y * (a * Math.exp(b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.1) or not (y <= 1.5e+27):
		tmp = x * ((math.pow(z, y) / a) / y)
	else:
		tmp = x / (y * (a * math.exp(b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.1) || !(y <= 1.5e+27))
		tmp = Float64(x * Float64(Float64((z ^ y) / a) / y));
	else
		tmp = Float64(x / Float64(y * Float64(a * exp(b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.1) || ~((y <= 1.5e+27)))
		tmp = x * (((z ^ y) / a) / y);
	else
		tmp = x / (y * (a * exp(b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.1], N[Not[LessEqual[y, 1.5e+27]], $MachinePrecision]], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(a * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.10000000000000009 or 1.49999999999999988e27 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-sum 79.5%

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

      4. associate-/l* 79.5%

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

      5. associate-/r/ 79.5%

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

      6. exp-neg 79.5%

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

      7. associate-*r/ 79.5%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in t around 0 65.7%

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

      1. *-commutative 65.7%

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

      2. associate-*l* 65.7%

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

      3. *-commutative 65.7%

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

   6. Simplified 65.7%

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

   7. Taylor expanded in b around 0 69.1%

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

      1. *-un-lft-identity 69.1%

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

      2. times-frac 81.4%

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

   9. Applied egg-rr 81.4%

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

       1. associate-*l/ 81.4%

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

       2. *-un-lft-identity 81.4%

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

   11. Applied egg-rr 81.4%

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

   if -2.10000000000000009 < y < 1.49999999999999988e27

   1. Initial program 95.7%

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

      1. associate-*r/ 95.6%

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

      2. sub-neg 95.6%

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

      3. exp-sum 77.7%

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

      4. associate-/l* 77.7%

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

      5. associate-/r/ 76.2%

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

      6. exp-neg 76.2%

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

      7. associate-*r/ 76.2%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in t around 0 65.9%

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

      1. *-commutative 65.9%

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

      2. associate-*l* 65.9%

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

      3. *-commutative 65.9%

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

   6. Simplified 65.9%

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

   7. Taylor expanded in y around 0 67.0%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \lor \neg \left(y \leq 1.5 \cdot 10^{+27}\right):\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \]

Alternative 7: 60.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot e^{b}\\ \mathbf{if}\;a \leq 5 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{x}{t_1}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (exp b))))
   (if (<= a 5e-171) (/ (/ x t_1) y) (/ x (* y t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * exp(b);
	double tmp;
	if (a <= 5e-171) {
		tmp = (x / t_1) / y;
	} else {
		tmp = x / (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 = a * exp(b)
    if (a <= 5d-171) then
        tmp = (x / t_1) / y
    else
        tmp = x / (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 = a * Math.exp(b);
	double tmp;
	if (a <= 5e-171) {
		tmp = (x / t_1) / y;
	} else {
		tmp = x / (y * t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * math.exp(b)
	tmp = 0
	if a <= 5e-171:
		tmp = (x / t_1) / y
	else:
		tmp = x / (y * t_1)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * exp(b))
	tmp = 0.0
	if (a <= 5e-171)
		tmp = Float64(Float64(x / t_1) / y);
	else
		tmp = Float64(x / Float64(y * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * exp(b);
	tmp = 0.0;
	if (a <= 5e-171)
		tmp = (x / t_1) / y;
	else
		tmp = x / (y * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[Exp[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 5e-171], N[(N[(x / t$95$1), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(y * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 4.99999999999999992e-171

   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. Taylor expanded in y around 0 82.7%

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

      1. exp-diff 68.4%

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

      2. sub-neg 68.4%

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

      3. metadata-eval 68.4%

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

      4. *-commutative 68.4%

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

      5. exp-to-pow 68.9%

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

   4. Simplified 68.9%

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

   5. Taylor expanded in t around 0 67.8%

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

   if 4.99999999999999992e-171 < a

   1. Initial program 97.1%

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

      1. associate-*r/ 98.4%

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

      2. sub-neg 98.4%

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

      3. exp-sum 79.6%

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

      4. associate-/l* 79.6%

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

      5. associate-/r/ 79.6%

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

      6. exp-neg 79.6%

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

      7. associate-*r/ 79.6%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in t around 0 64.0%

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

      1. *-commutative 64.0%

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

      2. associate-*l* 64.0%

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

      3. *-commutative 64.0%

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

   6. Simplified 64.0%

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

   7. Taylor expanded in y around 0 55.1%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \]

Alternative 8: 59.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.8%

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

   1. associate-*r/ 97.7%

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

   2. sub-neg 97.7%

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

   3. exp-sum 78.6%

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

   4. associate-/l* 78.6%

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

   5. associate-/r/ 77.8%

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

   6. exp-neg 77.8%

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

   7. associate-*r/ 77.8%

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

3. Simplified 67.3%

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

4. Taylor expanded in t around 0 65.8%

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

   1. *-commutative 65.8%

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

   2. associate-*l* 65.8%

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

   3. *-commutative 65.8%

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

6. Simplified 65.8%

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

7. Taylor expanded in y around 0 56.6%

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

8. Final simplification 56.6%

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

Alternative 9: 42.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y \cdot a}\\ \mathbf{if}\;b \leq -9 \cdot 10^{-6}:\\ \;\;\;\;t_1 - \left(\frac{b}{y} \cdot \frac{x}{a} + \left(b \cdot b\right) \cdot \left(t_1 \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* y a))))
   (if (<= b -9e-6)
     (- t_1 (+ (* (/ b y) (/ x a)) (* (* b b) (* t_1 -0.5))))
     (/ x (* a (* y (+ 1.0 b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * a);
	double tmp;
	if (b <= -9e-6) {
		tmp = t_1 - (((b / y) * (x / a)) + ((b * b) * (t_1 * -0.5)));
	} else {
		tmp = x / (a * (y * (1.0 + b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (y * a)
    if (b <= (-9d-6)) then
        tmp = t_1 - (((b / y) * (x / a)) + ((b * b) * (t_1 * (-0.5d0))))
    else
        tmp = x / (a * (y * (1.0d0 + b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * a);
	double tmp;
	if (b <= -9e-6) {
		tmp = t_1 - (((b / y) * (x / a)) + ((b * b) * (t_1 * -0.5)));
	} else {
		tmp = x / (a * (y * (1.0 + b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (y * a)
	tmp = 0
	if b <= -9e-6:
		tmp = t_1 - (((b / y) * (x / a)) + ((b * b) * (t_1 * -0.5)))
	else:
		tmp = x / (a * (y * (1.0 + b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(y * a))
	tmp = 0.0
	if (b <= -9e-6)
		tmp = Float64(t_1 - Float64(Float64(Float64(b / y) * Float64(x / a)) + Float64(Float64(b * b) * Float64(t_1 * -0.5))));
	else
		tmp = Float64(x / Float64(a * Float64(y * Float64(1.0 + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (y * a);
	tmp = 0.0;
	if (b <= -9e-6)
		tmp = t_1 - (((b / y) * (x / a)) + ((b * b) * (t_1 * -0.5)));
	else
		tmp = x / (a * (y * (1.0 + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -9e-6], N[(t$95$1 - N[(N[(N[(b / y), $MachinePrecision] * N[(x / a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -9.00000000000000023e-6

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-sum 66.0%

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

      4. associate-/l* 66.0%

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

      5. associate-/r/ 66.0%

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

      6. exp-neg 66.0%

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

      7. associate-*r/ 66.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in t around 0 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-*l* 70.3%

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

      3. *-commutative 70.3%

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

   6. Simplified 70.3%

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

   7. Taylor expanded in y around 0 76.4%

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

   8. Taylor expanded in b around 0 27.0%

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

      1. *-commutative 27.0%

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

      2. distribute-lft-out 27.0%

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

      3. times-frac 29.0%

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

      4. *-commutative 29.0%

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

      5. unpow2 29.0%

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

      6. distribute-rgt-out 49.1%

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

      7. metadata-eval 49.1%

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

      8. *-commutative 49.1%

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

      9. *-commutative 49.1%

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

   10. Simplified 49.1%

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

   if -9.00000000000000023e-6 < b

   1. Initial program 97.2%

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

      1. associate-*r/ 97.2%

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

      2. sub-neg 97.2%

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

      3. exp-sum 81.6%

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

      4. associate-/l* 81.6%

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

      5. associate-/r/ 80.6%

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

      6. exp-neg 80.6%

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

      7. associate-*r/ 80.6%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in t around 0 64.8%

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

      1. *-commutative 64.8%

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

      2. associate-*l* 64.8%

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

      3. *-commutative 64.8%

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

   6. Simplified 64.8%

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

   7. Taylor expanded in b around 0 59.8%

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

   8. Taylor expanded in y around 0 39.7%

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

   9. Taylor expanded in a around 0 39.8%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{y \cdot a} - \left(\frac{b}{y} \cdot \frac{x}{a} + \left(b \cdot b\right) \cdot \left(\frac{x}{y \cdot a} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}\\ \end{array} \]

Alternative 10: 39.4% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{x}{a} - \frac{x \cdot b}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.35e-170)
   (/ (- (/ x a) (/ (* x b) a)) y)
   (/ x (* a (* y (+ 1.0 b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.35e-170) {
		tmp = ((x / a) - ((x * b) / a)) / y;
	} else {
		tmp = x / (a * (y * (1.0 + b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.35d-170)) then
        tmp = ((x / a) - ((x * b) / a)) / y
    else
        tmp = x / (a * (y * (1.0d0 + b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.35e-170) {
		tmp = ((x / a) - ((x * b) / a)) / y;
	} else {
		tmp = x / (a * (y * (1.0 + b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1.35e-170:
		tmp = ((x / a) - ((x * b) / a)) / y
	else:
		tmp = x / (a * (y * (1.0 + b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.35e-170)
		tmp = Float64(Float64(Float64(x / a) - Float64(Float64(x * b) / a)) / y);
	else
		tmp = Float64(x / Float64(a * Float64(y * Float64(1.0 + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.35e-170)
		tmp = ((x / a) - ((x * b) / a)) / y;
	else
		tmp = x / (a * (y * (1.0 + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.35e-170], N[(N[(N[(x / a), $MachinePrecision] - N[(N[(x * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(a * N[(y * N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3499999999999999e-170

   1. Initial program 98.5%

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

      1. associate-*r/ 98.5%

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

      2. sub-neg 98.5%

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

      3. exp-sum 77.5%

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

      4. associate-/l* 77.5%

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

      5. associate-/r/ 77.5%

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

      6. exp-neg 77.5%

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

      7. associate-*r/ 77.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in t around 0 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-*l* 66.0%

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

      3. *-commutative 66.0%

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

   6. Simplified 66.0%

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

   7. Taylor expanded in y around 0 65.7%

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

   8. Taylor expanded in b around 0 38.3%

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

      1. mul-1-neg 38.3%

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

      2. unsub-neg 38.3%

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

      3. *-commutative 38.3%

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

      4. times-frac 41.9%

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

   10. Simplified 41.9%

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

   11. Taylor expanded in y around 0 46.0%

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

   if -1.3499999999999999e-170 < b

   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. Step-by-step derivation

      1. associate-*r/ 97.3%

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

      2. sub-neg 97.3%

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

      3. exp-sum 79.0%

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

      4. associate-/l* 79.0%

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

      5. associate-/r/ 77.9%

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

      6. exp-neg 77.9%

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

      7. associate-*r/ 77.9%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in t around 0 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-*l* 65.8%

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

      3. *-commutative 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in b around 0 59.9%

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

   8. Taylor expanded in y around 0 38.1%

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

   9. Taylor expanded in a around 0 38.2%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{x}{a} - \frac{x \cdot b}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}\\ \end{array} \]

Alternative 11: 37.6% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{b}{y} \cdot \frac{-x}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.95e-5)
   (* (/ b y) (/ (- x) a))
   (if (<= b 3.4e-80) (/ x (* y a)) (/ x (* y (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.95e-5) {
		tmp = (b / y) * (-x / a);
	} else if (b <= 3.4e-80) {
		tmp = x / (y * a);
	} else {
		tmp = x / (y * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.95d-5)) then
        tmp = (b / y) * (-x / a)
    else if (b <= 3.4d-80) then
        tmp = x / (y * a)
    else
        tmp = x / (y * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.95e-5) {
		tmp = (b / y) * (-x / a);
	} else if (b <= 3.4e-80) {
		tmp = x / (y * a);
	} else {
		tmp = x / (y * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1.95e-5:
		tmp = (b / y) * (-x / a)
	elif b <= 3.4e-80:
		tmp = x / (y * a)
	else:
		tmp = x / (y * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.95e-5)
		tmp = Float64(Float64(b / y) * Float64(Float64(-x) / a));
	elseif (b <= 3.4e-80)
		tmp = Float64(x / Float64(y * a));
	else
		tmp = Float64(x / Float64(y * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.95e-5)
		tmp = (b / y) * (-x / a);
	elseif (b <= 3.4e-80)
		tmp = x / (y * a);
	else
		tmp = x / (y * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.95e-5], N[(N[(b / y), $MachinePrecision] * N[((-x) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.4e-80], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.95e-5

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-sum 66.0%

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

      4. associate-/l* 66.0%

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

      5. associate-/r/ 66.0%

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

      6. exp-neg 66.0%

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

      7. associate-*r/ 66.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in t around 0 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-*l* 70.3%

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

      3. *-commutative 70.3%

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

   6. Simplified 70.3%

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

   7. Taylor expanded in y around 0 76.4%

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

   8. Taylor expanded in b around 0 32.1%

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

      1. mul-1-neg 32.1%

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

      2. unsub-neg 32.1%

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

      3. *-commutative 32.1%

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

      4. times-frac 37.9%

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

   10. Simplified 37.9%

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

   11. Taylor expanded in b around inf 32.1%

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

       1. times-frac 37.9%

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

       2. neg-mul-1 37.9%

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

       3. distribute-rgt-neg-out 37.9%

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

       4. distribute-neg-frac 37.9%

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

   13. Simplified 37.9%

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

   if -1.95e-5 < b < 3.4000000000000001e-80

   1. Initial program 95.3%

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

      1. associate-*r/ 95.2%

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

      2. sub-neg 95.2%

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

      3. exp-sum 95.2%

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

      4. associate-/l* 95.2%

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

      5. associate-/r/ 95.2%

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

      6. exp-neg 95.2%

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

      7. associate-*r/ 95.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in t around 0 67.3%

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

      1. *-commutative 67.3%

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

      2. associate-*l* 67.3%

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

      3. *-commutative 67.3%

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

   6. Simplified 67.3%

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

   7. Taylor expanded in y around 0 43.5%

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

   8. Taylor expanded in b around 0 43.2%

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

   if 3.4000000000000001e-80 < b

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-sum 61.9%

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

      4. associate-/l* 61.9%

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

      5. associate-/r/ 59.5%

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

      6. exp-neg 59.5%

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

      7. associate-*r/ 59.5%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in t around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-*l* 61.1%

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

      3. *-commutative 61.1%

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

   6. Simplified 61.1%

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

   7. Taylor expanded in b around 0 48.8%

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

   8. Taylor expanded in y around 0 34.2%

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

   9. Taylor expanded in b around inf 34.4%

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

       1. associate-*r* 32.1%

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

       2. *-commutative 32.1%

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

       3. associate-*r* 35.4%

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

   11. Simplified 35.4%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{b}{y} \cdot \frac{-x}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 12: 38.3% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05:\\ \;\;\;\;x \cdot \frac{-b}{y \cdot a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.05)
   (* x (/ (- b) (* y a)))
   (if (<= b 8.6e-79) (/ x (* y a)) (/ x (* y (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.05) {
		tmp = x * (-b / (y * a));
	} else if (b <= 8.6e-79) {
		tmp = x / (y * a);
	} else {
		tmp = x / (y * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.05d0)) then
        tmp = x * (-b / (y * a))
    else if (b <= 8.6d-79) then
        tmp = x / (y * a)
    else
        tmp = x / (y * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.05) {
		tmp = x * (-b / (y * a));
	} else if (b <= 8.6e-79) {
		tmp = x / (y * a);
	} else {
		tmp = x / (y * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1.05:
		tmp = x * (-b / (y * a))
	elif b <= 8.6e-79:
		tmp = x / (y * a)
	else:
		tmp = x / (y * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.05)
		tmp = Float64(x * Float64(Float64(-b) / Float64(y * a)));
	elseif (b <= 8.6e-79)
		tmp = Float64(x / Float64(y * a));
	else
		tmp = Float64(x / Float64(y * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.05)
		tmp = x * (-b / (y * a));
	elseif (b <= 8.6e-79)
		tmp = x / (y * a);
	else
		tmp = x / (y * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.05], N[(x * N[((-b) / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-79], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05000000000000004

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-sum 65.3%

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

      4. associate-/l* 65.3%

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

      5. associate-/r/ 65.3%

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

      6. exp-neg 65.3%

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

      7. associate-*r/ 65.3%

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

   3. Simplified 65.3%

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

   4. Taylor expanded in t around 0 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*l* 71.7%

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

      3. *-commutative 71.7%

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

   6. Simplified 71.7%

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

   7. Taylor expanded in y around 0 77.9%

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

   8. Taylor expanded in b around 0 32.6%

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

      1. mul-1-neg 32.6%

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

      2. unsub-neg 32.6%

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

      3. *-commutative 32.6%

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

      4. times-frac 36.6%

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

   10. Simplified 36.6%

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

   11. Taylor expanded in b around inf 32.6%

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

       1. times-frac 36.6%

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

       2. neg-mul-1 36.6%

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

       3. associate-*r/ 36.9%

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

       4. associate-/l* 36.6%

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

       5. associate-/r/ 44.7%

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

       6. associate-/r* 44.5%

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

       7. distribute-rgt-neg-in 44.5%

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

   13. Simplified 44.5%

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

   if -1.05000000000000004 < b < 8.59999999999999963e-79

   1. Initial program 95.3%

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

      1. associate-*r/ 95.2%

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

      2. sub-neg 95.2%

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

      3. exp-sum 95.2%

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

      4. associate-/l* 95.2%

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

      5. associate-/r/ 95.2%

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

      6. exp-neg 95.2%

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

      7. associate-*r/ 95.2%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in t around 0 66.8%

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

      1. *-commutative 66.8%

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

      2. associate-*l* 66.8%

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

      3. *-commutative 66.8%

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

   6. Simplified 66.8%

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

   7. Taylor expanded in y around 0 43.1%

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

   8. Taylor expanded in b around 0 42.9%

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

   if 8.59999999999999963e-79 < b

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-sum 61.9%

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

      4. associate-/l* 61.9%

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

      5. associate-/r/ 59.5%

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

      6. exp-neg 59.5%

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

      7. associate-*r/ 59.5%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in t around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-*l* 61.1%

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

      3. *-commutative 61.1%

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

   6. Simplified 61.1%

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

   7. Taylor expanded in b around 0 48.8%

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

   8. Taylor expanded in y around 0 34.2%

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

   9. Taylor expanded in b around inf 34.4%

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

       1. associate-*r* 32.1%

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

       2. *-commutative 32.1%

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

       3. associate-*r* 35.4%

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

   11. Simplified 35.4%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05:\\ \;\;\;\;x \cdot \frac{-b}{y \cdot a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 13: 39.4% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.95:\\ \;\;\;\;x \cdot \frac{-b}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -0.95) (* x (/ (- b) (* y a))) (/ x (* a (* y (+ 1.0 b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -0.95) {
		tmp = x * (-b / (y * a));
	} else {
		tmp = x / (a * (y * (1.0 + b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.95d0)) then
        tmp = x * (-b / (y * a))
    else
        tmp = x / (a * (y * (1.0d0 + b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -0.95) {
		tmp = x * (-b / (y * a));
	} else {
		tmp = x / (a * (y * (1.0 + b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -0.95:
		tmp = x * (-b / (y * a))
	else:
		tmp = x / (a * (y * (1.0 + b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -0.95)
		tmp = Float64(x * Float64(Float64(-b) / Float64(y * a)));
	else
		tmp = Float64(x / Float64(a * Float64(y * Float64(1.0 + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -0.95)
		tmp = x * (-b / (y * a));
	else
		tmp = x / (a * (y * (1.0 + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -0.95], N[(x * N[((-b) / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.94999999999999996

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-sum 65.3%

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

      4. associate-/l* 65.3%

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

      5. associate-/r/ 65.3%

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

      6. exp-neg 65.3%

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

      7. associate-*r/ 65.3%

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

   3. Simplified 65.3%

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

   4. Taylor expanded in t around 0 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*l* 71.7%

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

      3. *-commutative 71.7%

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

   6. Simplified 71.7%

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

   7. Taylor expanded in y around 0 77.9%

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

   8. Taylor expanded in b around 0 32.6%

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

      1. mul-1-neg 32.6%

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

      2. unsub-neg 32.6%

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

      3. *-commutative 32.6%

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

      4. times-frac 36.6%

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

   10. Simplified 36.6%

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

   11. Taylor expanded in b around inf 32.6%

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

       1. times-frac 36.6%

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

       2. neg-mul-1 36.6%

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

       3. associate-*r/ 36.9%

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

       4. associate-/l* 36.6%

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

       5. associate-/r/ 44.7%

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

       6. associate-/r* 44.5%

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

       7. distribute-rgt-neg-in 44.5%

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

   13. Simplified 44.5%

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

   if -0.94999999999999996 < b

   1. Initial program 97.2%

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

      1. associate-*r/ 97.2%

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

      2. sub-neg 97.2%

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

      3. exp-sum 81.7%

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

      4. associate-/l* 81.7%

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

      5. associate-/r/ 80.7%

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

      6. exp-neg 80.7%

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

      7. associate-*r/ 80.7%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in t around 0 64.5%

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

      1. *-commutative 64.5%

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

      2. associate-*l* 64.5%

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

      3. *-commutative 64.5%

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

   6. Simplified 64.5%

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

   7. Taylor expanded in b around 0 59.5%

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

   8. Taylor expanded in y around 0 39.5%

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

   9. Taylor expanded in a around 0 39.6%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.95:\\ \;\;\;\;x \cdot \frac{-b}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}\\ \end{array} \]

Alternative 14: 34.7% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 7.2e-79) (* x (/ 1.0 (* y a))) (/ x (* a (* y b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 7.2e-79) {
		tmp = x * (1.0 / (y * a));
	} else {
		tmp = x / (a * (y * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 7.2d-79) then
        tmp = x * (1.0d0 / (y * a))
    else
        tmp = x / (a * (y * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 7.2e-79) {
		tmp = x * (1.0 / (y * a));
	} else {
		tmp = x / (a * (y * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= 7.2e-79:
		tmp = x * (1.0 / (y * a))
	else:
		tmp = x / (a * (y * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 7.2e-79)
		tmp = Float64(x * Float64(1.0 / Float64(y * a)));
	else
		tmp = Float64(x / Float64(a * Float64(y * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= 7.2e-79)
		tmp = x * (1.0 / (y * a));
	else
		tmp = x / (a * (y * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 7.2e-79], N[(x * N[(1.0 / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.2000000000000005e-79

   1. Initial program 96.7%

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

      1. associate-*r/ 96.6%

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

      2. sub-neg 96.6%

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

      3. exp-sum 86.7%

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

      4. associate-/l* 86.7%

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

      5. associate-/r/ 86.7%

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

      6. exp-neg 86.7%

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

      7. associate-*r/ 86.7%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in t around 0 68.2%

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

      1. *-commutative 68.2%

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

      2. associate-*l* 68.2%

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

      3. *-commutative 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in b around 0 59.6%

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

   8. Taylor expanded in y around 0 38.0%

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

   if 7.2000000000000005e-79 < b

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-sum 61.9%

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

      4. associate-/l* 61.9%

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

      5. associate-/r/ 59.5%

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

      6. exp-neg 59.5%

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

      7. associate-*r/ 59.5%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in t around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-*l* 61.1%

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

      3. *-commutative 61.1%

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

   6. Simplified 61.1%

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

   7. Taylor expanded in b around 0 48.8%

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

   8. Taylor expanded in y around 0 34.2%

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

   9. Taylor expanded in b around inf 34.4%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 15: 34.8% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 5e-79) (* x (/ 1.0 (* y a))) (/ x (* y (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 5e-79) {
		tmp = x * (1.0 / (y * a));
	} else {
		tmp = x / (y * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 5d-79) then
        tmp = x * (1.0d0 / (y * a))
    else
        tmp = x / (y * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 5e-79) {
		tmp = x * (1.0 / (y * a));
	} else {
		tmp = x / (y * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= 5e-79:
		tmp = x * (1.0 / (y * a))
	else:
		tmp = x / (y * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 5e-79)
		tmp = Float64(x * Float64(1.0 / Float64(y * a)));
	else
		tmp = Float64(x / Float64(y * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= 5e-79)
		tmp = x * (1.0 / (y * a));
	else
		tmp = x / (y * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 5e-79], N[(x * N[(1.0 / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.99999999999999999e-79

   1. Initial program 96.7%

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

      1. associate-*r/ 96.6%

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

      2. sub-neg 96.6%

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

      3. exp-sum 86.7%

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

      4. associate-/l* 86.7%

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

      5. associate-/r/ 86.7%

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

      6. exp-neg 86.7%

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

      7. associate-*r/ 86.7%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in t around 0 68.2%

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

      1. *-commutative 68.2%

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

      2. associate-*l* 68.2%

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

      3. *-commutative 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in b around 0 59.6%

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

   8. Taylor expanded in y around 0 38.0%

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

   if 4.99999999999999999e-79 < b

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-sum 61.9%

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

      4. associate-/l* 61.9%

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

      5. associate-/r/ 59.5%

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

      6. exp-neg 59.5%

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

      7. associate-*r/ 59.5%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in t around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-*l* 61.1%

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

      3. *-commutative 61.1%

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

   6. Simplified 61.1%

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

   7. Taylor expanded in b around 0 48.8%

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

   8. Taylor expanded in y around 0 34.2%

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

   9. Taylor expanded in b around inf 34.4%

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

       1. associate-*r* 32.1%

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

       2. *-commutative 32.1%

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

       3. associate-*r* 35.4%

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

   11. Simplified 35.4%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 16: 30.6% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1}{y \cdot a} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* x (/ 1.0 (* y a))))

C

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

Fortran

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

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * (1.0 / (y * a));
}

Python

def code(x, y, z, t, a, b):
	return x * (1.0 / (y * a))

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[(1.0 / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.8%

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

   1. associate-*r/ 97.7%

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

   2. sub-neg 97.7%

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

   3. exp-sum 78.6%

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

   4. associate-/l* 78.6%

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

   5. associate-/r/ 77.8%

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

   6. exp-neg 77.8%

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

   7. associate-*r/ 77.8%

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

3. Simplified 67.3%

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

4. Taylor expanded in t around 0 65.8%

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

   1. *-commutative 65.8%

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

   2. associate-*l* 65.8%

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

   3. *-commutative 65.8%

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

6. Simplified 65.8%

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

7. Taylor expanded in b around 0 55.3%

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

8. Taylor expanded in y around 0 34.0%

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

9. Final simplification 34.0%

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

Alternative 17: 30.4% accurate, 63.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x / (y * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.8%

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

   1. associate-*r/ 97.7%

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

   2. sub-neg 97.7%

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

   3. exp-sum 78.6%

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

   4. associate-/l* 78.6%

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

   5. associate-/r/ 77.8%

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

   6. exp-neg 77.8%

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

   7. associate-*r/ 77.8%

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

3. Simplified 67.3%

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

4. Taylor expanded in t around 0 65.8%

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

   1. *-commutative 65.8%

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

   2. associate-*l* 65.8%

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

   3. *-commutative 65.8%

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

6. Simplified 65.8%

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

7. Taylor expanded in y around 0 56.6%

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

8. Taylor expanded in b around 0 33.8%

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

9. Final simplification 33.8%

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

Developer target: 71.8% 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 2023200 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

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