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

Percentage Accurate: 84.8% → 99.7%

Time: 11.3s

Alternatives: 7

Speedup: 9.7×

• could not determine a ground truth

  1. x = 1.276563805906115e+215
  2. y = -1.166280625204922e+41
  3. z = -8.806503010425338e-89

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: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{e^{-z}}{y}\\ \mathbf{if}\;y \leq -1.96:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ (exp (- z)) y))))
   (if (<= y -1.96) t_0 (if (<= y 1.95e-8) (+ 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 <= -1.96) {
		tmp = t_0;
	} else if (y <= 1.95e-8) {
		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 <= (-1.96d0)) then
        tmp = t_0
    else if (y <= 1.95d-8) 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 <= -1.96) {
		tmp = t_0;
	} else if (y <= 1.95e-8) {
		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 <= -1.96:
		tmp = t_0
	elif y <= 1.95e-8:
		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 <= -1.96)
		tmp = t_0;
	elseif (y <= 1.95e-8)
		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 <= -1.96)
		tmp = t_0;
	elseif (y <= 1.95e-8)
		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, -1.96], t$95$0, If[LessEqual[y, 1.95e-8], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.96 or 1.94999999999999992e-8 < y

   1. Initial program 85.0%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 -1.96 < y < 1.94999999999999992e-8

   1. Initial program 78.8%

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

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

   5. Simplified 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.96:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+186}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, 1\right)}{y}\\ \mathbf{elif}\;z \leq -920:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.05e+186)
   (/ (fma y x 1.0) y)
   (if (<= z -920.0) (/ (exp (- z)) y) (+ x (/ 1.0 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.05e+186) {
		tmp = fma(y, x, 1.0) / y;
	} else if (z <= -920.0) {
		tmp = exp(-z) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.05e+186)
		tmp = Float64(fma(y, x, 1.0) / y);
	elseif (z <= -920.0)
		tmp = Float64(exp(Float64(-z)) / y);
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.05e+186], N[(N[(y * x + 1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, -920.0], N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.05e186

   1. Initial program 41.1%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 58.5

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

   5. Simplified 58.5%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 75.6

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

   8. Simplified 75.6%

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

   if -2.05e186 < z < -920

   1. Initial program 44.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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-lowering-neg.f64 73.0

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

   5. Simplified 73.0%

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

   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-lowering-neg.f64 73.0

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

   8. Simplified 73.0%

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

   if -920 < z

   1. Initial program 93.3%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 96.1

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

   5. Simplified 96.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+186}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, 1\right)}{y}\\ \mathbf{elif}\;z \leq -920:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, 1\right)}{y}\\ \mathbf{elif}\;z \leq -1450:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, x \cdot x, x\right), 1\right) - z}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6e+96)
   (/ (fma y x 1.0) y)
   (if (<= z -1450.0)
     (/ (- (fma y (fma y (* x x) x) 1.0) z) y)
     (+ x (/ 1.0 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6e+96) {
		tmp = fma(y, x, 1.0) / y;
	} else if (z <= -1450.0) {
		tmp = (fma(y, fma(y, (x * x), x), 1.0) - z) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6e+96)
		tmp = Float64(fma(y, x, 1.0) / y);
	elseif (z <= -1450.0)
		tmp = Float64(Float64(fma(y, fma(y, Float64(x * x), x), 1.0) - z) / y);
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6e+96], N[(N[(y * x + 1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, -1450.0], N[(N[(N[(y * N[(y * N[(x * x), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.0000000000000001e96

   1. Initial program 50.7%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 50.2

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

   5. Simplified 50.2%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 59.0

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

   8. Simplified 59.0%

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

   if -6.0000000000000001e96 < z < -1450

   1. Initial program 34.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. /-lowering-/.f64 11.3

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

   5. Simplified 11.3%

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

      1. associate-+r- N/A

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

      2. flip-+ N/A

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

      3. frac-sub N/A

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

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

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

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

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

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

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

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

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. Applied egg-rr 10.4%

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

   8. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. --lowering--.f64 19.1

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

   10. Simplified 19.1%

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

   11. Taylor expanded in y around 0

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

   13. Simplified 52.1%

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

   if -1450 < z

   1. Initial program 93.3%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 96.1

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

   5. Simplified 96.1%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, 1\right)}{y}\\ \mathbf{elif}\;z \leq -1450:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, x \cdot x, x\right), 1\right) - z}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+70}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.15e+70) x (if (<= y 7e-69) (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.15e+70) {
		tmp = x;
	} else if (y <= 7e-69) {
		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 (y <= (-1.15d+70)) then
        tmp = x
    else if (y <= 7d-69) 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 (y <= -1.15e+70) {
		tmp = x;
	} else if (y <= 7e-69) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.15e+70:
		tmp = x
	elif y <= 7e-69:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.15e+70)
		tmp = x;
	elseif (y <= 7e-69)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.15e+70)
		tmp = x;
	elseif (y <= 7e-69)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.15e+70], x, If[LessEqual[y, 7e-69], N[(1.0 / y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1.14999999999999997e70 or 7.0000000000000003e-69 < y

   1. Initial program 84.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 67.5%

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

   if -1.14999999999999997e70 < y < 7.0000000000000003e-69

   1. Initial program 80.1%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 71.0

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

   5. Simplified 71.0%

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

Alternative 5: 85.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.2e+37) (/ (fma y x 1.0) y) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.2e+37) {
		tmp = fma(y, x, 1.0) / y;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.2e+37)
		tmp = Float64(fma(y, x, 1.0) / y);
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.2e+37], N[(N[(y * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1999999999999998e37

   1. Initial program 46.5%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 41.5

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

   5. Simplified 41.5%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 52.3

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

   8. Simplified 52.3%

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

   if -5.1999999999999998e37 < z

   1. Initial program 90.0%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 92.2

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

   5. Simplified 92.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.2% 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 82.7%

   \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
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 83.7

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

5. Simplified 83.7%

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

6. Final simplification 83.7%

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

Alternative 7: 49.6% 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 82.7%

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

3. Taylor expanded in x around inf

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

5. Simplified 49.5%

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

Developer Target 1: 91.2% 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 2024199 
(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)))