Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.4% → 96.6%

Time: 6.5s

Alternatives: 7

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: 84.4% 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: 96.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-78) (fma (/ y z) x x) (fma (/ x z) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-78) {
		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 <= 2e-78)
		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, 2e-78], 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 < 2e-78

   1. Initial program 82.2%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if 2e-78 < y

   1. Initial program 89.3%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. div-add N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 60.5

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y \cdot z + {z}^{2}}{{z}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto x \cdot \frac{y \cdot z + \color{blue}{z \cdot z}}{{z}^{2}} \]

      3. distribute-rgt-out N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. *-inverses N/A

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

      7. associate-*l* N/A

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

      8. *-rgt-identity N/A

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

      9. div-add N/A

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

      10. distribute-lft-in N/A

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

      11. *-inverses N/A

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

      12. associate-/l* N/A

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

      13. associate-*l/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 98.6

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

   7. Applied rewrites 98.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 50.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (+ y z)) z)))
   (if (<= t_0 0.0)
     (* (/ y z) x)
     (if (<= t_0 2e+296) (/ (* z x) z) (* z (/ x z))))))

C

double code(double x, double y, double z) {
	double t_0 = (x * (y + z)) / z;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (y / z) * x;
	} else if (t_0 <= 2e+296) {
		tmp = (z * x) / z;
	} else {
		tmp = z * (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) :: t_0
    real(8) :: tmp
    t_0 = (x * (y + z)) / z
    if (t_0 <= 0.0d0) then
        tmp = (y / z) * x
    else if (t_0 <= 2d+296) then
        tmp = (z * x) / z
    else
        tmp = z * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * (y + z)) / z;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (y / z) * x;
	} else if (t_0 <= 2e+296) {
		tmp = (z * x) / z;
	} else {
		tmp = z * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * (y + z)) / z
	tmp = 0
	if t_0 <= 0.0:
		tmp = (y / z) * x
	elif t_0 <= 2e+296:
		tmp = (z * x) / z
	else:
		tmp = z * (x / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * Float64(y + z)) / z)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(y / z) * x);
	elseif (t_0 <= 2e+296)
		tmp = Float64(Float64(z * x) / z);
	else
		tmp = Float64(z * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * (y + z)) / z;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (y / z) * x;
	elseif (t_0 <= 2e+296)
		tmp = (z * x) / z;
	else
		tmp = z * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2e+296], N[(N[(z * x), $MachinePrecision] / z), $MachinePrecision], N[(z * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < 0.0

   1. Initial program 82.1%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   7. 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 50.3

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

   8. Applied rewrites 50.3%

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

   10. Applied rewrites 48.5%

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

   if 0.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < 1.99999999999999996e296

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 1.99999999999999996e296 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 61.7%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 82.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 60.0%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 42.5%

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

Alternative 3: 65.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-126} \lor \neg \left(y \leq 7.8 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.5e-126) (not (<= y 7.8e+48))) (* (/ x z) y) (* z (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.5e-126) || !(y <= 7.8e+48)) {
		tmp = (x / z) * y;
	} else {
		tmp = z * (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 ((y <= (-4.5d-126)) .or. (.not. (y <= 7.8d+48))) then
        tmp = (x / z) * y
    else
        tmp = z * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.5e-126) || !(y <= 7.8e+48)) {
		tmp = (x / z) * y;
	} else {
		tmp = z * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.5e-126) or not (y <= 7.8e+48):
		tmp = (x / z) * y
	else:
		tmp = z * (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.5e-126) || !(y <= 7.8e+48))
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = Float64(z * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.5e-126) || ~((y <= 7.8e+48)))
		tmp = (x / z) * y;
	else
		tmp = z * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.5e-126], N[Not[LessEqual[y, 7.8e+48]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], N[(z * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.50000000000000025e-126 or 7.8000000000000002e48 < y

   1. Initial program 88.9%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   7. 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 69.6

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

   8. Applied rewrites 69.6%

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

   if -4.50000000000000025e-126 < y < 7.8000000000000002e48

   1. Initial program 79.1%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 15.3%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 69.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-126} \lor \neg \left(y \leq 7.8 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+48}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-78)
   (/ (* y x) z)
   (if (<= y 7.8e+48) (* z (/ x z)) (* (/ x z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-78) {
		tmp = (y * x) / z;
	} else if (y <= 7.8e+48) {
		tmp = z * (x / z);
	} 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 (y <= (-2d-78)) then
        tmp = (y * x) / z
    else if (y <= 7.8d+48) then
        tmp = z * (x / z)
    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 (y <= -2e-78) {
		tmp = (y * x) / z;
	} else if (y <= 7.8e+48) {
		tmp = z * (x / z);
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2e-78:
		tmp = (y * x) / z
	elif y <= 7.8e+48:
		tmp = z * (x / z)
	else:
		tmp = (x / z) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-78)
		tmp = Float64(Float64(y * x) / z);
	elseif (y <= 7.8e+48)
		tmp = Float64(z * Float64(x / z));
	else
		tmp = Float64(Float64(x / z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e-78)
		tmp = (y * x) / z;
	elseif (y <= 7.8e+48)
		tmp = z * (x / z);
	else
		tmp = (x / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-78], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 7.8e+48], N[(z * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e-78

   1. Initial program 88.3%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 72.9

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

   5. Applied rewrites 72.9%

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

   if -2e-78 < y < 7.8000000000000002e48

   1. Initial program 78.6%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 15.6%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 67.7%

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

   if 7.8000000000000002e48 < y

   1. Initial program 92.3%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   7. 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 70.9

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

   8. Applied rewrites 70.9%

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

Alternative 5: 64.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-126}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+48}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-126)
   (* (/ y z) x)
   (if (<= y 7.8e+48) (* z (/ x z)) (* (/ x z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-126) {
		tmp = (y / z) * x;
	} else if (y <= 7.8e+48) {
		tmp = z * (x / z);
	} 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 (y <= (-4d-126)) then
        tmp = (y / z) * x
    else if (y <= 7.8d+48) then
        tmp = z * (x / z)
    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 (y <= -4e-126) {
		tmp = (y / z) * x;
	} else if (y <= 7.8e+48) {
		tmp = z * (x / z);
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4e-126:
		tmp = (y / z) * x
	elif y <= 7.8e+48:
		tmp = z * (x / z)
	else:
		tmp = (x / z) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e-126)
		tmp = Float64(Float64(y / z) * x);
	elseif (y <= 7.8e+48)
		tmp = Float64(z * Float64(x / z));
	else
		tmp = Float64(Float64(x / z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4e-126)
		tmp = (y / z) * x;
	elseif (y <= 7.8e+48)
		tmp = z * (x / z);
	else
		tmp = (x / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e-126], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 7.8e+48], N[(z * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9999999999999998e-126

   1. Initial program 86.9%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   7. 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 68.9

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

   8. Applied rewrites 68.9%

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

   10. Applied rewrites 69.4%

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

   if -3.9999999999999998e-126 < y < 7.8000000000000002e48

   1. Initial program 79.1%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 15.3%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 69.3%

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

   if 7.8000000000000002e48 < y

   1. Initial program 92.3%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   7. 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 70.9

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

   8. Applied rewrites 70.9%

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

Alternative 6: 96.0% 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 84.4%

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

Alternative 7: 43.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

7. Applied rewrites 76.9%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 36.7%

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

11. Taylor expanded in y around 0

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

13. Applied rewrites 43.5%

    \[\leadsto z \cdot \frac{x}{\color{blue}{z}} \]
14. Add Preprocessing

Developer Target 1: 96.3% 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 2024329 
(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))