Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.6% → 95.6%

Time: 6.0s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

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 * (y + z)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (y + z)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\frac{z + y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e+147) (* (/ (+ z y) z) x) (/ (* x (+ y z)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+147) {
		tmp = ((z + y) / z) * x;
	} else {
		tmp = (x * (y + z)) / z;
	}
	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 <= 5d+147) then
        tmp = ((z + y) / z) * x
    else
        tmp = (x * (y + z)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+147) {
		tmp = ((z + y) / z) * x;
	} else {
		tmp = (x * (y + z)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5e+147:
		tmp = ((z + y) / z) * x
	else:
		tmp = (x * (y + z)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e+147)
		tmp = Float64(Float64(Float64(z + y) / z) * x);
	else
		tmp = Float64(Float64(x * Float64(y + z)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e+147)
		tmp = ((z + y) / z) * x;
	else
		tmp = (x * (y + z)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e+147], N[(N[(N[(z + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000002e147

   1. Initial program 82.4%

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

   4. Applied rewrites 98.6%

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

   if 5.0000000000000002e147 < y

   1. Initial program 94.5%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 70.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+139}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-293}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.2e+139)
   (* 1.0 x)
   (if (<= z 1.7e-293)
     (* (/ y z) x)
     (if (<= z 1.85e-14) (* (/ x z) y) (* 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2e+139) {
		tmp = 1.0 * x;
	} else if (z <= 1.7e-293) {
		tmp = (y / z) * x;
	} else if (z <= 1.85e-14) {
		tmp = (x / z) * y;
	} else {
		tmp = 1.0 * 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 <= (-2.2d+139)) then
        tmp = 1.0d0 * x
    else if (z <= 1.7d-293) then
        tmp = (y / z) * x
    else if (z <= 1.85d-14) then
        tmp = (x / z) * y
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2e+139) {
		tmp = 1.0 * x;
	} else if (z <= 1.7e-293) {
		tmp = (y / z) * x;
	} else if (z <= 1.85e-14) {
		tmp = (x / z) * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.2e+139:
		tmp = 1.0 * x
	elif z <= 1.7e-293:
		tmp = (y / z) * x
	elif z <= 1.85e-14:
		tmp = (x / z) * y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.2e+139)
		tmp = Float64(1.0 * x);
	elseif (z <= 1.7e-293)
		tmp = Float64(Float64(y / z) * x);
	elseif (z <= 1.85e-14)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.2e+139)
		tmp = 1.0 * x;
	elseif (z <= 1.7e-293)
		tmp = (y / z) * x;
	elseif (z <= 1.85e-14)
		tmp = (x / z) * y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.2e+139], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, 1.7e-293], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.85e-14], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1999999999999999e139 or 1.85000000000000001e-14 < z

   1. Initial program 71.3%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 77.0%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -2.1999999999999999e139 < z < 1.7e-293

   1. Initial program 92.0%

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 21.7%

      \[\leadsto \color{blue}{1} \cdot x \]

   8. Taylor expanded in y around inf

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

      1. lower-/.f64 77.0

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

   10. Applied rewrites 77.0%

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

   if 1.7e-293 < z < 1.85000000000000001e-14

   1. Initial program 95.3%

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

   4. Applied rewrites 87.4%

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

   5. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 82.2

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

   7. Applied rewrites 82.2%

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

Alternative 3: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+139} \lor \neg \left(z \leq 1.85 \cdot 10^{-14}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.15e+139) (not (<= z 1.85e-14))) (* 1.0 x) (/ (* y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.15e+139) || !(z <= 1.85e-14)) {
		tmp = 1.0 * x;
	} else {
		tmp = (y * x) / z;
	}
	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 <= (-2.15d+139)) .or. (.not. (z <= 1.85d-14))) then
        tmp = 1.0d0 * x
    else
        tmp = (y * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.15e+139) || !(z <= 1.85e-14)) {
		tmp = 1.0 * x;
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.15e+139) or not (z <= 1.85e-14):
		tmp = 1.0 * x
	else:
		tmp = (y * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.15e+139) || !(z <= 1.85e-14))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(y * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.15e+139) || ~((z <= 1.85e-14)))
		tmp = 1.0 * x;
	else
		tmp = (y * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.15e+139], N[Not[LessEqual[z, 1.85e-14]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1499999999999999e139 or 1.85000000000000001e-14 < z

   1. Initial program 71.3%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 77.0%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -2.1499999999999999e139 < z < 1.85000000000000001e-14

   1. Initial program 93.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+139} \lor \neg \left(z \leq 1.85 \cdot 10^{-14}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+139} \lor \neg \left(z \leq 1.85 \cdot 10^{-14}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.15e+139) (not (<= z 1.85e-14))) (* 1.0 x) (* (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.15e+139) || !(z <= 1.85e-14)) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.15d+139)) .or. (.not. (z <= 1.85d-14))) then
        tmp = 1.0d0 * x
    else
        tmp = (x / z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.15e+139) || !(z <= 1.85e-14)) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.15e+139) or not (z <= 1.85e-14):
		tmp = 1.0 * x
	else:
		tmp = (x / z) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.15e+139) || !(z <= 1.85e-14))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(x / z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.15e+139) || ~((z <= 1.85e-14)))
		tmp = 1.0 * x;
	else
		tmp = (x / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.15e+139], N[Not[LessEqual[z, 1.85e-14]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1499999999999999e139 or 1.85000000000000001e-14 < z

   1. Initial program 71.3%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 77.0%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -2.1499999999999999e139 < z < 1.85000000000000001e-14

   1. Initial program 93.4%

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 76.1

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

   7. Applied rewrites 76.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+139} \lor \neg \left(z \leq 1.85 \cdot 10^{-14}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+128}:\\ \;\;\;\;\frac{z + y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+128) (* (/ (+ z y) z) x) (fma (/ x z) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+128) {
		tmp = ((z + y) / z) * x;
	} else {
		tmp = fma((x / z), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e+128)
		tmp = Float64(Float64(Float64(z + y) / z) * x);
	else
		tmp = fma(Float64(x / z), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e+128], N[(N[(N[(z + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0000000000000001e128

   1. Initial program 82.9%

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

   4. Applied rewrites 98.6%

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

   if 1.0000000000000001e128 < y

   1. Initial program 90.3%

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

   4. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. lift-/.f64 N/A

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

      7. *-inverses N/A

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

      8. distribute-lft1-in N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lower-fma.f64 93.4

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

   6. Applied rewrites 93.4%

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

Alternative 6: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+128) (fma (/ y z) x x) (fma (/ x z) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+128) {
		tmp = fma((y / z), x, x);
	} else {
		tmp = fma((x / z), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e+128)
		tmp = fma(Float64(y / z), x, x);
	else
		tmp = fma(Float64(x / z), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e+128], N[(N[(y / z), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0000000000000001e128

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. *-inverses N/A

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

      6. +-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 1.0000000000000001e128 < y

   1. Initial program 90.3%

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

   4. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. lift-/.f64 N/A

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

      7. *-inverses N/A

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

      8. distribute-lft1-in N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lower-fma.f64 93.4

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

   6. Applied rewrites 93.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{z}, x, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

   5. *-inverses N/A

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

   6. +-commutative N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 96.2

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

5. Applied rewrites 96.2%

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

6. Final simplification 96.2%

   \[\leadsto \mathsf{fma}\left(\frac{y}{z}, x, x\right) \]
7. Add Preprocessing

Alternative 8: 50.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return 1.0 * 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 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

4. Applied rewrites 96.2%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 44.0%

   \[\leadsto \color{blue}{1} \cdot x \]
8. Add Preprocessing

Developer Target 1: 96.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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 / (z / (y + z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024339 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (/ x (/ z (+ y z))))

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