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

Percentage Accurate: 85.6% → 99.7%

Time: 9.2s

Alternatives: 10

Speedup: 8.7×

• could not determine a ground truth

  1. x = -7.392186030505984e-168
  2. y = 1.2078812502162624e+38
  3. z = -7.019809514471782e-279

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: 85.6% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2:\\ \;\;\;\;\frac{1}{e^{z} \cdot y} + x\\ \mathbf{elif}\;y \leq 0.05:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.2)
   (+ (/ 1.0 (* (exp z) y)) x)
   (if (<= y 0.05) (+ (/ 1.0 y) x) (+ (/ (exp (- z)) y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.2) {
		tmp = (1.0 / (exp(z) * y)) + x;
	} else if (y <= 0.05) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = (exp(-z) / y) + 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 <= (-1.2d0)) then
        tmp = (1.0d0 / (exp(z) * y)) + x
    else if (y <= 0.05d0) then
        tmp = (1.0d0 / y) + x
    else
        tmp = (exp(-z) / y) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.2) {
		tmp = (1.0 / (Math.exp(z) * y)) + x;
	} else if (y <= 0.05) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = (Math.exp(-z) / y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.2:
		tmp = (1.0 / (math.exp(z) * y)) + x
	elif y <= 0.05:
		tmp = (1.0 / y) + x
	else:
		tmp = (math.exp(-z) / y) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.2)
		tmp = Float64(Float64(1.0 / Float64(exp(z) * y)) + x);
	elseif (y <= 0.05)
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = Float64(Float64(exp(Float64(-z)) / y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.2)
		tmp = (1.0 / (exp(z) * y)) + x;
	elseif (y <= 0.05)
		tmp = (1.0 / y) + x;
	else
		tmp = (exp(-z) / y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.2], N[(N[(1.0 / N[(N[Exp[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 0.05], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.19999999999999996

   1. Initial program 85.9%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-exp.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -1.19999999999999996 < y < 0.050000000000000003

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.4%

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

   if 0.050000000000000003 < y

   1. Initial program 81.6%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto x + \frac{e^{\color{blue}{-z}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2:\\ \;\;\;\;\frac{1}{e^{z} \cdot y} + x\\ \mathbf{elif}\;y \leq 0.05:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-z}}{y} + x\\ \mathbf{if}\;y \leq -1.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.05:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ (exp (- z)) y) x)))
   (if (<= y -1.2) t_0 (if (<= y 0.05) (+ (/ 1.0 y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (exp(-z) / y) + x;
	double tmp;
	if (y <= -1.2) {
		tmp = t_0;
	} else if (y <= 0.05) {
		tmp = (1.0 / y) + x;
	} 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 = (exp(-z) / y) + x
    if (y <= (-1.2d0)) then
        tmp = t_0
    else if (y <= 0.05d0) then
        tmp = (1.0d0 / y) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.exp(-z) / y) + x;
	double tmp;
	if (y <= -1.2) {
		tmp = t_0;
	} else if (y <= 0.05) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.exp(-z) / y) + x
	tmp = 0
	if y <= -1.2:
		tmp = t_0
	elif y <= 0.05:
		tmp = (1.0 / y) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(exp(Float64(-z)) / y) + x)
	tmp = 0.0
	if (y <= -1.2)
		tmp = t_0;
	elseif (y <= 0.05)
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (exp(-z) / y) + x;
	tmp = 0.0;
	if (y <= -1.2)
		tmp = t_0;
	elseif (y <= 0.05)
		tmp = (1.0 / y) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.2], t$95$0, If[LessEqual[y, 0.05], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.19999999999999996 or 0.050000000000000003 < y

   1. Initial program 83.8%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.19999999999999996 < y < 0.050000000000000003

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.4%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2:\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \mathbf{elif}\;y \leq 0.05:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z, 0.5\right), z, -1\right), z, 1\right)}{y} + x\\ \mathbf{elif}\;y \leq 0.05:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, 0.5, y\right), z, y\right)} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.05)
   (+ (/ (fma (fma (fma -0.16666666666666666 z 0.5) z -1.0) z 1.0) y) x)
   (if (<= y 0.05)
     (+ (/ 1.0 y) x)
     (+ (/ 1.0 (fma (fma (* z y) 0.5 y) z y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.05) {
		tmp = (fma(fma(fma(-0.16666666666666666, z, 0.5), z, -1.0), z, 1.0) / y) + x;
	} else if (y <= 0.05) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = (1.0 / fma(fma((z * y), 0.5, y), z, y)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.05)
		tmp = Float64(Float64(fma(fma(fma(-0.16666666666666666, z, 0.5), z, -1.0), z, 1.0) / y) + x);
	elseif (y <= 0.05)
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = Float64(Float64(1.0 / fma(fma(Float64(z * y), 0.5, y), z, y)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.05], N[(N[(N[(N[(N[(-0.16666666666666666 * z + 0.5), $MachinePrecision] * z + -1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 0.05], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(z * y), $MachinePrecision] * 0.5 + y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05000000000000004

   1. Initial program 85.9%

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

   3. Taylor expanded in z around 0

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

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 86.0%

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

   if -1.05000000000000004 < y < 0.050000000000000003

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.4%

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

   if 0.050000000000000003 < y

   1. Initial program 81.6%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-exp.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 87.0%

       \[\leadsto x + \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, 0.5, y\right), \color{blue}{z}, y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z, 0.5\right), z, -1\right), z, 1\right)}{y} + x\\ \mathbf{elif}\;y \leq 0.05:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, 0.5, y\right), z, y\right)} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), z, 1\right)}{y} + x\\ \mathbf{elif}\;y \leq 0.05:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, 0.5, y\right), z, y\right)} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.05)
   (+ (/ (fma (fma 0.5 z -1.0) z 1.0) y) x)
   (if (<= y 0.05)
     (+ (/ 1.0 y) x)
     (+ (/ 1.0 (fma (fma (* z y) 0.5 y) z y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.05) {
		tmp = (fma(fma(0.5, z, -1.0), z, 1.0) / y) + x;
	} else if (y <= 0.05) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = (1.0 / fma(fma((z * y), 0.5, y), z, y)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.05)
		tmp = Float64(Float64(fma(fma(0.5, z, -1.0), z, 1.0) / y) + x);
	elseif (y <= 0.05)
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = Float64(Float64(1.0 / fma(fma(Float64(z * y), 0.5, y), z, y)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.05], N[(N[(N[(N[(0.5 * z + -1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 0.05], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(z * y), $MachinePrecision] * 0.5 + y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05000000000000004

   1. Initial program 85.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 83.4%

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

   if -1.05000000000000004 < y < 0.050000000000000003

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.4%

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

   if 0.050000000000000003 < y

   1. Initial program 81.6%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-exp.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 87.0%

       \[\leadsto x + \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, 0.5, y\right), \color{blue}{z}, y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), z, 1\right)}{y} + x\\ \mathbf{elif}\;y \leq 0.05:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, 0.5, y\right), z, y\right)} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), z, 1\right)}{y} + x\\ \mathbf{elif}\;y \leq 0.05:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, y, y\right)} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.05)
   (+ (/ (fma (fma 0.5 z -1.0) z 1.0) y) x)
   (if (<= y 0.05) (+ (/ 1.0 y) x) (+ (/ 1.0 (fma z y y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.05) {
		tmp = (fma(fma(0.5, z, -1.0), z, 1.0) / y) + x;
	} else if (y <= 0.05) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = (1.0 / fma(z, y, y)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.05)
		tmp = Float64(Float64(fma(fma(0.5, z, -1.0), z, 1.0) / y) + x);
	elseif (y <= 0.05)
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = Float64(Float64(1.0 / fma(z, y, y)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.05], N[(N[(N[(N[(0.5 * z + -1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 0.05], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 / N[(z * y + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05000000000000004

   1. Initial program 85.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 83.4%

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

   if -1.05000000000000004 < y < 0.050000000000000003

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.4%

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

   if 0.050000000000000003 < y

   1. Initial program 81.6%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 85.7

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

   10. Applied rewrites 85.7%

       \[\leadsto x + \frac{1}{\color{blue}{\mathsf{fma}\left(z, y, y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, -1\right), z, 1\right)}{y} + x\\ \mathbf{elif}\;y \leq 0.05:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, y, y\right)} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+160}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot 0.5}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.22e+160) (+ (/ (* (* z z) 0.5) y) x) (+ (/ 1.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.22e+160) {
		tmp = (((z * z) * 0.5) / y) + x;
	} else {
		tmp = (1.0 / y) + 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 (z <= (-1.22d+160)) then
        tmp = (((z * z) * 0.5d0) / y) + x
    else
        tmp = (1.0d0 / y) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.22e+160) {
		tmp = (((z * z) * 0.5) / y) + x;
	} else {
		tmp = (1.0 / y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.22e+160:
		tmp = (((z * z) * 0.5) / y) + x
	else:
		tmp = (1.0 / y) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.22e+160)
		tmp = Float64(Float64(Float64(Float64(z * z) * 0.5) / y) + x);
	else
		tmp = Float64(Float64(1.0 / y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.22e+160)
		tmp = (((z * z) * 0.5) / y) + x;
	else
		tmp = (1.0 / y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.22e+160], N[(N[(N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.22e160

   1. Initial program 67.0%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 37.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 73.6%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + \frac{1}{2} \cdot \frac{{z}^{2}}{\color{blue}{y}} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.3%

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

   12. Taylor expanded in z around inf

       \[\leadsto x + \frac{1}{2} \cdot \frac{{z}^{2}}{\color{blue}{y}} \]
   13. Step-by-step derivation

   14. Applied rewrites 73.6%

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

   if -1.22e160 < z

   1. Initial program 89.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 89.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+160}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot 0.5}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.05:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, y, y\right)} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.05) (+ (/ 1.0 y) x) (+ (/ 1.0 (fma z y y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.05) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = (1.0 / fma(z, y, y)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.05)
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = Float64(Float64(1.0 / fma(z, y, y)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.05], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 / N[(z * y + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.050000000000000003

   1. Initial program 89.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 84.2%

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

   if 0.050000000000000003 < y

   1. Initial program 81.6%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 85.7

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

   10. Applied rewrites 85.7%

       \[\leadsto x + \frac{1}{\color{blue}{\mathsf{fma}\left(z, y, y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.05:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, y, y\right)} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.6% accurate, 15.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (1.0 / y) + 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 = (1.0d0 / y) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 82.7%

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

6. Final simplification 82.7%

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

Alternative 9: 40.0% accurate, 19.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = 1.0d0 / y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 38.4

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

5. Applied rewrites 38.4%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
6. Add Preprocessing

Alternative 10: 2.2% accurate, 19.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return -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 = (-1.0d0) / y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 38.4

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

5. Applied rewrites 38.4%

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

7. Applied rewrites 10.5%

   \[\leadsto {\left(y \cdot y\right)}^{\color{blue}{-0.5}} \]

8. Taylor expanded in y around -inf

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

10. Applied rewrites 2.0%

    \[\leadsto \frac{-1}{\color{blue}{y}} \]
11. Add Preprocessing

Developer Target 1: 91.8% 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 2024244 
(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)))