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

Percentage Accurate: 84.8% → 99.2%

Time: 11.3s

Alternatives: 7

Speedup: 7.1×

• could not determine a ground truth

  1. x = 1.2692059972384285e+268
  2. y = -2.8662104095623516e+34
  3. z = -2.092193619730886e-63

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: 84.8% 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: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{e^{0 - z}}{y}\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.1:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ (exp (- 0.0 z)) y))))
   (if (<= y -2.5e+20) t_0 (if (<= y 0.1) (+ 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 <= -2.5e+20) {
		tmp = t_0;
	} else if (y <= 0.1) {
		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 <= (-2.5d+20)) then
        tmp = t_0
    else if (y <= 0.1d0) 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 <= -2.5e+20) {
		tmp = t_0;
	} else if (y <= 0.1) {
		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 <= -2.5e+20:
		tmp = t_0
	elif y <= 0.1:
		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 <= -2.5e+20)
		tmp = t_0;
	elseif (y <= 0.1)
		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 <= -2.5e+20)
		tmp = t_0;
	elseif (y <= 0.1)
		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, -2.5e+20], t$95$0, If[LessEqual[y, 0.1], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5e20 or 0.10000000000000001 < y

   1. Initial program 84.5%

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

   3. Taylor expanded in y around inf

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

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

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 100.0

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

   5. Simplified 100.0%

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

   if -2.5e20 < y < 0.10000000000000001

   1. Initial program 80.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. /-lowering-/.f64 99.2

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

   5. Simplified 99.2%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+20}:\\ \;\;\;\;x + \frac{e^{0 - z}}{y}\\ \mathbf{elif}\;y \leq 0.1:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{0 - z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -28500000:\\ \;\;\;\;\frac{e^{0 - z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -28500000.0) (/ (exp (- 0.0 z)) y) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -28500000.0) {
		tmp = exp((0.0 - z)) / 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 <= (-28500000.0d0)) then
        tmp = exp((0.0d0 - z)) / 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 <= -28500000.0) {
		tmp = Math.exp((0.0 - z)) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -28500000.0:
		tmp = math.exp((0.0 - z)) / y
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -28500000.0)
		tmp = Float64(exp(Float64(0.0 - z)) / 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 <= -28500000.0)
		tmp = exp((0.0 - z)) / y;
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -28500000.0], N[(N[Exp[N[(0.0 - z), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.85e7

   1. Initial program 42.1%

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

   3. Taylor expanded in y around inf

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

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

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 63.3

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

   5. Simplified 63.3%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 63.3

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

   8. Simplified 63.3%

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

   if -2.85e7 < z

   1. Initial program 92.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. /-lowering-/.f64 95.6

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

   5. Simplified 95.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -28500000:\\ \;\;\;\;\frac{e^{0 - z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9.8e+131)
   (/ (* -0.16666666666666666 (* z (* z z))) y)
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.8e+131) {
		tmp = (-0.16666666666666666 * (z * (z * z))) / 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 <= (-9.8d+131)) then
        tmp = ((-0.16666666666666666d0) * (z * (z * z))) / 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 <= -9.8e+131) {
		tmp = (-0.16666666666666666 * (z * (z * z))) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9.8e+131:
		tmp = (-0.16666666666666666 * (z * (z * z))) / y
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9.8e+131)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(z * Float64(z * z))) / 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 <= -9.8e+131)
		tmp = (-0.16666666666666666 * (z * (z * z))) / y;
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9.8e+131], N[(N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.80000000000000064e131

   1. Initial program 38.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

   5. Simplified 65.5%

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.5}{y} + \mathsf{fma}\left(z, 0 - \left(0.16666666666666666 + \left(\frac{0.5}{y} + \frac{0.3333333333333333}{y \cdot y}\right)\right), 0.5\right), -1\right), 1\right)}}{y} \]

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. /-lowering-/.f64 52.6

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

   8. Simplified 52.6%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{{z}^{3}}{y}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot {z}^{3}}{y}} \]

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

          \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot {z}^{3}}{y}} \]

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

          \[\leadsto \frac{\color{blue}{\frac{-1}{6} \cdot {z}^{3}}}{y} \]

       4. cube-mult N/A

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 65.5

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

   11. Simplified 65.5%

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

   if -9.80000000000000064e131 < z

   1. Initial program 87.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. /-lowering-/.f64 89.8

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

   5. Simplified 89.8%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+20}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.5e+20) (+ x (/ (fma z (fma z 0.5 -1.0) 1.0) y)) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.5e+20) {
		tmp = x + (fma(z, fma(z, 0.5, -1.0), 1.0) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.5e+20)
		tmp = Float64(x + Float64(fma(z, fma(z, 0.5, -1.0), 1.0) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.5e+20], N[(x + N[(N[(z * N[(z * 0.5 + -1.0), $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 < -2.5e20

   1. Initial program 79.9%

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

   3. Taylor expanded in y around inf

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

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

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 77.3

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

   8. Simplified 77.3%

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

   if -2.5e20 < y

   1. Initial program 84.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. /-lowering-/.f64 91.7

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

   5. Simplified 91.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+20}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-162}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.45e-162) x (if (<= x 1.05e-68) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-162) {
		tmp = x;
	} else if (x <= 1.05e-68) {
		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 <= (-1.45d-162)) then
        tmp = x
    else if (x <= 1.05d-68) 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 <= -1.45e-162) {
		tmp = x;
	} else if (x <= 1.05e-68) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.45e-162:
		tmp = x
	elif x <= 1.05e-68:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.45e-162)
		tmp = x;
	elseif (x <= 1.05e-68)
		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 <= -1.45e-162)
		tmp = x;
	elseif (x <= 1.05e-68)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.45e-162], x, If[LessEqual[x, 1.05e-68], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4500000000000001e-162 or 1.05000000000000004e-68 < x

   1. Initial program 87.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 70.0%

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

   if -1.4500000000000001e-162 < x < 1.05000000000000004e-68

   1. Initial program 75.3%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 63.6

         \[\leadsto \color{blue}{\frac{1}{y}} \]

   5. Simplified 63.6%

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

Alternative 6: 84.5% accurate, 15.6× 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 82.9%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

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

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

   3. /-lowering-/.f64 85.3

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

5. Simplified 85.3%

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

6. Final simplification 85.3%

   \[\leadsto x + \frac{1}{y} \]
7. Add Preprocessing

Alternative 7: 50.0% accurate, 234.0× 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 82.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 54.0%

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

Developer Target 1: 91.1% 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 2024195 
(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)))