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

Percentage Accurate: 84.3% → 99.4%

Time: 10.6s

Alternatives: 6

Speedup: 7.1×

• could not determine a ground truth

  1. x = 8.305627830514385e-19
  2. y = 7.617048770160387e+39
  3. z = -1.1772904797094018e-104

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: 84.3% 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: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{e^{-z}}{y}\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ (exp (- z)) y))))
   (if (<= y -5.2e+14) t_0 (if (<= y 2e-9) (+ x (/ 1.0 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (exp(-z) / y);
	double tmp;
	if (y <= -5.2e+14) {
		tmp = t_0;
	} else if (y <= 2e-9) {
		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(-z) / y)
    if (y <= (-5.2d+14)) then
        tmp = t_0
    else if (y <= 2d-9) 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(-z) / y);
	double tmp;
	if (y <= -5.2e+14) {
		tmp = t_0;
	} else if (y <= 2e-9) {
		tmp = x + (1.0 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (math.exp(-z) / y)
	tmp = 0
	if y <= -5.2e+14:
		tmp = t_0
	elif y <= 2e-9:
		tmp = x + (1.0 / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(exp(Float64(-z)) / y))
	tmp = 0.0
	if (y <= -5.2e+14)
		tmp = t_0;
	elseif (y <= 2e-9)
		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(-z) / y);
	tmp = 0.0;
	if (y <= -5.2e+14)
		tmp = t_0;
	elseif (y <= 2e-9)
		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[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.2e+14], t$95$0, If[LessEqual[y, 2e-9], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2e14 or 2.00000000000000012e-9 < y

   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 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-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   if -5.2e14 < y < 2.00000000000000012e-9

   1. Initial program 86.7%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.2e+14)
   (+ x (/ (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.2e+14) {
		tmp = x + (fma(z, fma(z, fma(z, -0.16666666666666666, 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 <= -5.2e+14)
		tmp = Float64(x + Float64(fma(z, fma(z, fma(z, -0.16666666666666666, 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, -5.2e+14], N[(x + N[(N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -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 < -5.2e14

   1. Initial program 85.6%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 89.0%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 89.0

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

   8. Simplified 89.0%

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

   if -5.2e14 < y

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
   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 93.9

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

   5. Simplified 93.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.2e+14) (+ 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 <= -5.2e+14) {
		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 <= -5.2e+14)
		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, -5.2e+14], 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 < -5.2e14

   1. Initial program 85.6%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   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 83.1

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

   8. Simplified 83.1%

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

   if -5.2e14 < y

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1}{y}} \]
   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 93.9

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

   5. Simplified 93.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.1e-18) x (if (<= x 1.1e-34) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.1e-18) {
		tmp = x;
	} else if (x <= 1.1e-34) {
		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 <= (-1.1d-18)) then
        tmp = x
    else if (x <= 1.1d-34) 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 <= -1.1e-18) {
		tmp = x;
	} else if (x <= 1.1e-34) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.1e-18:
		tmp = x
	elif x <= 1.1e-34:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.1e-18)
		tmp = x;
	elseif (x <= 1.1e-34)
		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 <= -1.1e-18)
		tmp = x;
	elseif (x <= 1.1e-34)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.1e-18], x, If[LessEqual[x, 1.1e-34], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0999999999999999e-18 or 1.0999999999999999e-34 < x

   1. Initial program 86.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 70.4%

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

   if -1.0999999999999999e-18 < x < 1.0999999999999999e-34

   1. Initial program 87.5%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 69.4

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

   5. Simplified 69.4%

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

Alternative 5: 84.3% 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 86.9%

   \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
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 87.1

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

5. Simplified 87.1%

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

6. Final simplification 87.1%

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

Alternative 6: 48.7% 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 86.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 43.8%

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

Developer Target 1: 90.9% 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 2024198 
(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)))