Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G

Percentage Accurate: 84.9% → 99.7%

Time: 11.1s

Alternatives: 7

Speedup: 42.2×

Specification

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))

C

double code(double x, double y, double z) {
	return x + (exp((y * log((y / (z + y))))) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (exp((y * log((y / (z + y))))) / y)
end function

Java

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

Python

def code(x, y, z):
	return x + (math.exp((y * math.log((y / (z + y))))) / y)

Julia

function code(x, y, z)
	return Float64(x + Float64(exp(Float64(y * log(Float64(y / Float64(z + y))))) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (exp((y * log((y / (z + y))))) / y);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[Exp[N[(y * N[Log[N[(y / N[(z + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 84.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))

C

double code(double x, double y, double z) {
	return x + (exp((y * log((y / (z + y))))) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (exp((y * log((y / (z + y))))) / y)
end function

Java

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

Python

def code(x, y, z):
	return x + (math.exp((y * math.log((y / (z + y))))) / y)

Julia

function code(x, y, z)
	return Float64(x + Float64(exp(Float64(y * log(Float64(y / Float64(z + y))))) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (exp((y * log((y / (z + y))))) / y);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[Exp[N[(y * N[Log[N[(y / N[(z + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{e^{0 - z}}{y}\\ \mathbf{if}\;y \leq -160:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ (exp (- 0.0 z)) y))))
   (if (<= y -160.0) t_0 (if (<= y 0.5) (+ x (/ 1.0 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (exp((0.0 - z)) / y);
	double tmp;
	if (y <= -160.0) {
		tmp = t_0;
	} else if (y <= 0.5) {
		tmp = x + (1.0 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (exp((0.0d0 - z)) / y)
    if (y <= (-160.0d0)) then
        tmp = t_0
    else if (y <= 0.5d0) then
        tmp = x + (1.0d0 / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (Math.exp((0.0 - z)) / y);
	double tmp;
	if (y <= -160.0) {
		tmp = t_0;
	} else if (y <= 0.5) {
		tmp = x + (1.0 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (math.exp((0.0 - z)) / y)
	tmp = 0
	if y <= -160.0:
		tmp = t_0
	elif y <= 0.5:
		tmp = x + (1.0 / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(exp(Float64(0.0 - z)) / y))
	tmp = 0.0
	if (y <= -160.0)
		tmp = t_0;
	elseif (y <= 0.5)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (exp((0.0 - z)) / y);
	tmp = 0.0;
	if (y <= -160.0)
		tmp = t_0;
	elseif (y <= 0.5)
		tmp = x + (1.0 / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(N[Exp[N[(0.0 - z), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -160.0], t$95$0, If[LessEqual[y, 0.5], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -160 or 0.5 < y

   1. Initial program 86.9%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 86.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 86.9%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{-1 \cdot z}}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{-1 \cdot z}\right), \color{blue}{y}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot z\right)\right), y\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right), y\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - z\right)\right), y\right)\right) \]

      6. --lowering--.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right), y\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{e^{0 - z}}{y}} \]

   if -160 < y < 0.5

   1. Initial program 83.9%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 83.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 83.9%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]

      2. /-lowering-/.f64 99.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

   7. Simplified 99.4%

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -160:\\ \;\;\;\;x \cdot \left(1 + \left(z \cdot \left(\frac{0.5}{y} + \frac{0.5}{y \cdot y}\right) + \frac{-1}{y}\right) \cdot \frac{z}{x}\right) + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -160.0)
   (+
    (*
     x
     (+ 1.0 (* (+ (* z (+ (/ 0.5 y) (/ 0.5 (* y y)))) (/ -1.0 y)) (/ z x))))
    (/ 1.0 y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -160.0) {
		tmp = (x * (1.0 + (((z * ((0.5 / y) + (0.5 / (y * y)))) + (-1.0 / y)) * (z / x)))) + (1.0 / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-160.0d0)) then
        tmp = (x * (1.0d0 + (((z * ((0.5d0 / y) + (0.5d0 / (y * y)))) + ((-1.0d0) / y)) * (z / x)))) + (1.0d0 / y)
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -160.0) {
		tmp = (x * (1.0 + (((z * ((0.5 / y) + (0.5 / (y * y)))) + (-1.0 / y)) * (z / x)))) + (1.0 / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -160.0:
		tmp = (x * (1.0 + (((z * ((0.5 / y) + (0.5 / (y * y)))) + (-1.0 / y)) * (z / x)))) + (1.0 / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -160.0)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(Float64(Float64(z * Float64(Float64(0.5 / y) + Float64(0.5 / Float64(y * y)))) + Float64(-1.0 / y)) * Float64(z / x)))) + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -160.0)
		tmp = (x * (1.0 + (((z * ((0.5 / y) + (0.5 / (y * y)))) + (-1.0 / y)) * (z / x)))) + (1.0 / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -160.0], N[(N[(x * N[(1.0 + N[(N[(N[(z * N[(N[(0.5 / y), $MachinePrecision] + N[(0.5 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -160

   1. Initial program 88.5%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 88.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 88.5%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{y}\right), \color{blue}{\left(x + z \cdot \left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{y}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \left(\color{blue}{x} + z \cdot \left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{y}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{y}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{y}\right)}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)}\right)\right)\right)\right) \]

   7. Simplified 77.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \color{blue}{\left(x \cdot \left(1 + \frac{z \cdot \left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{y}\right)}{x}\right)\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{z \cdot \left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{y}\right)}{x}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{y}\right)}{x}\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{y}\right) \cdot z}{x}\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{y}\right) \cdot \color{blue}{\frac{z}{x}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{y}\right), \color{blue}{\left(\frac{z}{x}\right)}\right)\right)\right)\right) \]

   10. Simplified 81.1%

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

   if -160 < y

   1. Initial program 84.5%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 84.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]

      2. /-lowering-/.f64 92.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

   7. Simplified 92.0%

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -160:\\ \;\;\;\;x \cdot \left(1 + \left(z \cdot \left(\frac{0.5}{y} + \frac{0.5}{y \cdot y}\right) + \frac{-1}{y}\right) \cdot \frac{z}{x}\right) + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -160:\\ \;\;\;\;x + \frac{z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right) - -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -160.0)
   (+ x (/ (- (* z (+ -1.0 (* z (+ 0.5 (* z -0.16666666666666666))))) -1.0) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -160.0) {
		tmp = x + (((z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666))))) - -1.0) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-160.0d0)) then
        tmp = x + (((z * ((-1.0d0) + (z * (0.5d0 + (z * (-0.16666666666666666d0)))))) - (-1.0d0)) / y)
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -160.0) {
		tmp = x + (((z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666))))) - -1.0) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -160.0:
		tmp = x + (((z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666))))) - -1.0) / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -160.0)
		tmp = Float64(x + Float64(Float64(Float64(z * Float64(-1.0 + Float64(z * Float64(0.5 + Float64(z * -0.16666666666666666))))) - -1.0) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -160.0)
		tmp = x + (((z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666))))) - -1.0) / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -160.0], N[(x + N[(N[(N[(z * N[(-1.0 + N[(z * N[(0.5 + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -160

   1. Initial program 88.5%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 88.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 88.5%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

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

   7. Simplified 78.2%

      \[\leadsto \color{blue}{\left(x + \frac{1}{y}\right) + z \cdot \left(z \cdot \left(\frac{0.5}{y \cdot y} + \left(\frac{0.5}{y} - \frac{z \cdot \left(\frac{0.5}{y} + \left(0.16666666666666666 + \frac{0.3333333333333333}{y \cdot y}\right)\right)}{y}\right)\right) + \frac{-1}{y}\right)} \]

   8. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\frac{z \cdot \left(1 + -1 \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right)\right)\right) - 1}{y}\right)\right) \]

      2. unsub-neg N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{z \cdot \left(1 + -1 \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right)\right)\right) - 1}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(z \cdot \left(1 + -1 \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right)\right)\right) - 1\right), \color{blue}{y}\right)\right) \]

   10. Simplified 80.8%

       \[\leadsto \color{blue}{x - \frac{-1 + z \cdot \left(1 - z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)}{y}} \]

   if -160 < y

   1. Initial program 84.5%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 84.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]

      2. /-lowering-/.f64 92.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

   7. Simplified 92.0%

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -160:\\ \;\;\;\;x + \frac{z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right) - -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.1% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+127}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot \frac{-0.16666666666666666}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.2e+127)
   (* (* z (* z z)) (/ -0.16666666666666666 y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.2e+127) {
		tmp = (z * (z * z)) * (-0.16666666666666666 / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-8.2d+127)) then
        tmp = (z * (z * z)) * ((-0.16666666666666666d0) / y)
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.2e+127) {
		tmp = (z * (z * z)) * (-0.16666666666666666 / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8.2e+127:
		tmp = (z * (z * z)) * (-0.16666666666666666 / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.2e+127)
		tmp = Float64(Float64(z * Float64(z * z)) * Float64(-0.16666666666666666 / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8.2e+127)
		tmp = (z * (z * z)) * (-0.16666666666666666 / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.2e+127], N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.19999999999999965e127

   1. Initial program 54.0%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 54.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

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

   7. Simplified 43.2%

      \[\leadsto \color{blue}{\left(x + \frac{1}{y}\right) + z \cdot \left(z \cdot \left(\frac{0.5}{y \cdot y} + \left(\frac{0.5}{y} - \frac{z \cdot \left(\frac{0.5}{y} + \left(0.16666666666666666 + \frac{0.3333333333333333}{y \cdot y}\right)\right)}{y}\right)\right) + \frac{-1}{y}\right)} \]

   8. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(x + \frac{1}{y}\right), \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot z\right) - 1\right)}{y}\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{y} + x\right), \left(\frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot z\right) - 1\right)}}{y}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{y}\right), x\right), \left(\frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot z\right) - 1\right)}}{y}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \left(\frac{\color{blue}{z} \cdot \left(z \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot z\right) - 1\right)}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \left(\frac{\left(z \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot z\right) - 1\right) \cdot z}{y}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \left(\left(z \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot z\right) - 1\right) \cdot \color{blue}{\frac{z}{y}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \mathsf{*.f64}\left(\left(z \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot z\right) - 1\right), \color{blue}{\left(\frac{z}{y}\right)}\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \mathsf{*.f64}\left(\left(z \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\frac{\color{blue}{z}}{y}\right)\right)\right) \]

      10. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \mathsf{*.f64}\left(\left(z \cdot \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\frac{z}{y}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \mathsf{*.f64}\left(\left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\frac{z}{y}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \mathsf{*.f64}\left(\left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + -1\right), \left(\frac{z}{y}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \mathsf{*.f64}\left(\left(-1 + z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right)\right), \left(\frac{\color{blue}{z}}{y}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right)\right)\right), \left(\frac{\color{blue}{z}}{y}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right)\right)\right), \left(\frac{z}{y}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot z\right)\right)\right)\right), \left(\frac{z}{y}\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \left(z \cdot \frac{-1}{6}\right)\right)\right)\right), \left(\frac{z}{y}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right), \left(\frac{z}{y}\right)\right)\right) \]

      19. /-lowering-/.f64 54.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right), \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right)\right) \]

   10. Simplified 54.5%

       \[\leadsto \color{blue}{\left(\frac{1}{y} + x\right) + \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right) \cdot \frac{z}{y}} \]

   11. Taylor expanded in z around inf

       \[\leadsto \color{blue}{{z}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{y \cdot z} - \frac{1}{6} \cdot \frac{1}{y}\right)} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\left({z}^{3}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{y \cdot z} - \frac{1}{6} \cdot \frac{1}{y}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(z \cdot \left(z \cdot z\right)\right), \left(\color{blue}{\frac{1}{2} \cdot \frac{1}{y \cdot z}} - \frac{1}{6} \cdot \frac{1}{y}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(z \cdot {z}^{2}\right), \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{y \cdot z}} - \frac{1}{6} \cdot \frac{1}{y}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left({z}^{2}\right)\right), \left(\color{blue}{\frac{1}{2} \cdot \frac{1}{y \cdot z}} - \frac{1}{6} \cdot \frac{1}{y}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot z\right)\right), \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{y \cdot z}} - \frac{1}{6} \cdot \frac{1}{y}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{y \cdot z}} - \frac{1}{6} \cdot \frac{1}{y}\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \left(\frac{1}{2} \cdot \frac{1}{y \cdot z} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{y}\right)\right)}\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{y \cdot z} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot \frac{1}{y}}\right)\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \left(\frac{\frac{1}{2}}{y \cdot z} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6}} \cdot \frac{1}{y}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{y \cdot z}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{y}\right)\right)}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(y \cdot z\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot \frac{1}{y}}\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(z \cdot y\right)\right), \left(\mathsf{neg}\left(\frac{1}{6} \cdot \color{blue}{\frac{1}{y}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, y\right)\right), \left(\mathsf{neg}\left(\frac{1}{6} \cdot \color{blue}{\frac{1}{y}}\right)\right)\right)\right) \]

       14. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, y\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{6} \cdot 1}{y}\right)\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, y\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{6}}{y}\right)\right)\right)\right) \]

       16. distribute-neg-frac N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, y\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{6}\right)}{\color{blue}{y}}\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, y\right)\right), \left(\frac{\frac{-1}{6}}{y}\right)\right)\right) \]

       18. /-lowering-/.f64 61.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\frac{-1}{6}, \color{blue}{y}\right)\right)\right) \]

   13. Simplified 61.5%

       \[\leadsto \color{blue}{\left(z \cdot \left(z \cdot z\right)\right) \cdot \left(\frac{0.5}{z \cdot y} + \frac{-0.16666666666666666}{y}\right)} \]

   14. Taylor expanded in z around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \color{blue}{\left(\frac{\frac{-1}{6}}{y}\right)}\right) \]
   15. Step-by-step derivation

       1. /-lowering-/.f64 61.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{/.f64}\left(\frac{-1}{6}, \color{blue}{y}\right)\right) \]

   16. Simplified 61.5%

       \[\leadsto \left(z \cdot \left(z \cdot z\right)\right) \cdot \color{blue}{\frac{-0.16666666666666666}{y}} \]

   if -8.19999999999999965e127 < z

   1. Initial program 89.3%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 89.3%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 89.3%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]

      2. /-lowering-/.f64 92.1%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

   7. Simplified 92.1%

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 62.4% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+89}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.2e+89) x (if (<= x 6.6e-99) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.2e+89) {
		tmp = x;
	} else if (x <= 6.6e-99) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-8.2d+89)) then
        tmp = x
    else if (x <= 6.6d-99) then
        tmp = 1.0d0 / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.2e+89) {
		tmp = x;
	} else if (x <= 6.6e-99) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8.2e+89:
		tmp = x
	elif x <= 6.6e-99:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.2e+89)
		tmp = x;
	elseif (x <= 6.6e-99)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8.2e+89)
		tmp = x;
	elseif (x <= 6.6e-99)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.2e+89], x, If[LessEqual[x, 6.6e-99], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -8.1999999999999997e89 or 6.59999999999999973e-99 < x

   1. Initial program 90.8%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 90.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 90.8%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   6. Step-by-step derivation

   7. Simplified 74.8%

      \[\leadsto \color{blue}{x} \]

   if -8.1999999999999997e89 < x < 6.59999999999999973e-99

   1. Initial program 80.9%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

      4. exp-to-pow N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

      8. +-lowering-+.f64 80.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

   3. Simplified 80.9%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{y}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 68.7%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{y}\right) \]

   7. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{1}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 85.7% accurate, 42.2× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (+ x (/ 1.0 y)))

C

double code(double x, double y, double z) {
	return x + (1.0 / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (1.0d0 / y)
end function

Java

public static double code(double x, double y, double z) {
	return x + (1.0 / y);
}

Python

def code(x, y, z):
	return x + (1.0 / y)

Julia

function code(x, y, z)
	return Float64(x + Float64(1.0 / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (1.0 / y);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.6%

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

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

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

   4. exp-to-pow N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

   8. +-lowering-+.f64 85.6%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

3. Simplified 85.6%

   \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
4. Add Preprocessing

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + \frac{1}{y}} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{y}\right)}\right) \]

   2. /-lowering-/.f64 86.8%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{y}\right)\right) \]

7. Simplified 86.8%

   \[\leadsto \color{blue}{x + \frac{1}{y}} \]
8. Add Preprocessing

Alternative 7: 49.2% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 85.6%

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

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

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{y \cdot \log \left(\frac{y}{z + y}\right)}\right), \color{blue}{y}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(e^{\log \left(\frac{y}{z + y}\right) \cdot y}\right), y\right)\right) \]

   4. exp-to-pow N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left({\left(\frac{y}{z + y}\right)}^{y}\right), y\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{z + y}\right), y\right), y\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(z + y\right)\right), y\right), y\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \left(y + z\right)\right), y\right), y\right)\right) \]

   8. +-lowering-+.f64 85.6%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, z\right)\right), y\right), y\right)\right) \]

3. Simplified 85.6%

   \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
6. Step-by-step derivation

7. Simplified 47.7%

   \[\leadsto \color{blue}{x} \]
8. Add Preprocessing

Developer Target 1: 91.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.11541576 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< (/ y (+ z y)) 7.11541576e-315)
   (+ x (/ (exp (/ -1.0 z)) y))
   (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y / (z + y)) < 7.11541576e-315) {
		tmp = x + (exp((-1.0 / z)) / y);
	} else {
		tmp = x + (exp(log(pow((y / (y + z)), y))) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y / (z + y)) < 7.11541576d-315) then
        tmp = x + (exp(((-1.0d0) / z)) / y)
    else
        tmp = x + (exp(log(((y / (y + z)) ** y))) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y / (z + y)) < 7.11541576e-315) {
		tmp = x + (Math.exp((-1.0 / z)) / y);
	} else {
		tmp = x + (Math.exp(Math.log(Math.pow((y / (y + z)), y))) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y / (z + y)) < 7.11541576e-315:
		tmp = x + (math.exp((-1.0 / z)) / y)
	else:
		tmp = x + (math.exp(math.log(math.pow((y / (y + z)), y))) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y / Float64(z + y)) < 7.11541576e-315)
		tmp = Float64(x + Float64(exp(Float64(-1.0 / z)) / y));
	else
		tmp = Float64(x + Float64(exp(log((Float64(y / Float64(y + z)) ^ y))) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y / (z + y)) < 7.11541576e-315)
		tmp = x + (exp((-1.0 / z)) / y);
	else
		tmp = x + (exp(log(((y / (y + z)) ^ y))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[N[(y / N[(z + y), $MachinePrecision]), $MachinePrecision], 7.11541576e-315], N[(x + N[(N[Exp[N[(-1.0 / z), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Exp[N[Log[N[Power[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024161 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ y (+ z y)) 17788539399477/2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y))))

  (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))