Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 85.0% → 96.8%

Time: 5.4s

Alternatives: 5

Speedup: 0.6×

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: 85.0% 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: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5 \cdot 10^{-225}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{z}, y, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{z + y}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= x_m 3.5e-225) (fma (/ x_m z) y x_m) (/ x_m (/ z (+ z y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 3.5e-225) {
		tmp = fma((x_m / z), y, x_m);
	} else {
		tmp = x_m / (z / (z + y));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 3.5e-225)
		tmp = fma(Float64(x_m / z), y, x_m);
	else
		tmp = Float64(x_m / Float64(z / Float64(z + y)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 3.5e-225], N[(N[(x$95$m / z), $MachinePrecision] * y + x$95$m), $MachinePrecision], N[(x$95$m / N[(z / N[(z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.4999999999999997e-225

   1. Initial program 83.8%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 94.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 94.2

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

   4. Applied rewrites 94.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. lft-mult-inverse N/A

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

      8. associate-/r/ N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-commutative N/A

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

      16. associate-*l/ N/A

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

      17. lift-/.f64 N/A

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

      18. *-lft-identity N/A

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

      19. lower-fma.f64 92.5

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

   6. Applied rewrites 92.5%

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

   if 3.4999999999999997e-225 < x

   1. Initial program 84.7%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.6

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 97.6

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

   4. Applied rewrites 97.6%

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

Alternative 2: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \left(y + z\right)}{z} \leq -5 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{z}, y, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x\_m, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (+ y z)) z) -5e+80)
    (fma (/ x_m z) y x_m)
    (fma (/ y z) x_m x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (y + z)) / z) <= -5e+80) {
		tmp = fma((x_m / z), y, x_m);
	} else {
		tmp = fma((y / z), x_m, x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(y + z)) / z) <= -5e+80)
		tmp = fma(Float64(x_m / z), y, x_m);
	else
		tmp = fma(Float64(y / z), x_m, x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -5e+80], N[(N[(x$95$m / z), $MachinePrecision] * y + x$95$m), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -4.99999999999999961e80

   1. Initial program 75.2%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 92.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 92.7

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

   4. Applied rewrites 92.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. lft-mult-inverse N/A

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

      8. associate-/r/ N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-commutative N/A

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

      16. associate-*l/ N/A

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

      17. lift-/.f64 N/A

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

      18. *-lft-identity N/A

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

      19. lower-fma.f64 95.4

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

   6. Applied rewrites 95.4%

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

   if -4.99999999999999961e80 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 87.0%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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. associate-/l* N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. *-inverses N/A

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

      9. *-inverses N/A

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

      10. distribute-frac-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-rgt-out-- N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. associate-*l/ N/A

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

      16. associate-*r/ N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. cancel-sign-sub N/A

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

   5. Applied rewrites 96.5%

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

Alternative 3: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \left(y + z\right)}{z} \leq -5 \cdot 10^{+80}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x\_m, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (+ y z)) z) -5e+80)
    (* (/ x_m z) y)
    (fma (/ y z) x_m x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (y + z)) / z) <= -5e+80) {
		tmp = (x_m / z) * y;
	} else {
		tmp = fma((y / z), x_m, x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(y + z)) / z) <= -5e+80)
		tmp = Float64(Float64(x_m / z) * y);
	else
		tmp = fma(Float64(y / z), x_m, x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -5e+80], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -4.99999999999999961e80

   1. Initial program 75.2%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 92.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 92.7

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

   4. Applied rewrites 92.7%

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

   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 59.6

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

   7. Applied rewrites 59.6%

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

   if -4.99999999999999961e80 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 87.0%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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. associate-/l* N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. *-inverses N/A

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

      9. *-inverses N/A

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

      10. distribute-frac-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-rgt-out-- N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. associate-*l/ N/A

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

      16. associate-*r/ N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. cancel-sign-sub N/A

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

   5. Applied rewrites 96.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -5 \cdot 10^{+80}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{+30} \lor \neg \left(z \leq 2 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{x\_m}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= z -1.08e+30) (not (<= z 2e-39))) (/ x_m 1.0) (* (/ x_m z) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.08e+30) || !(z <= 2e-39)) {
		tmp = x_m / 1.0;
	} else {
		tmp = (x_m / z) * y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.08d+30)) .or. (.not. (z <= 2d-39))) then
        tmp = x_m / 1.0d0
    else
        tmp = (x_m / z) * y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.08e+30) || !(z <= 2e-39)) {
		tmp = x_m / 1.0;
	} else {
		tmp = (x_m / z) * y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (z <= -1.08e+30) or not (z <= 2e-39):
		tmp = x_m / 1.0
	else:
		tmp = (x_m / z) * y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((z <= -1.08e+30) || !(z <= 2e-39))
		tmp = Float64(x_m / 1.0);
	else
		tmp = Float64(Float64(x_m / z) * y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z <= -1.08e+30) || ~((z <= 2e-39)))
		tmp = x_m / 1.0;
	else
		tmp = (x_m / z) * y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[z, -1.08e+30], N[Not[LessEqual[z, 2e-39]], $MachinePrecision]], N[(x$95$m / 1.0), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.08e30 or 1.99999999999999986e-39 < z

   1. Initial program 82.9%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 78.8%

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

   if -1.08e30 < z < 1.99999999999999986e-39

   1. Initial program 86.1%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 89.4

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 89.4

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

   4. Applied rewrites 89.4%

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

   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 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{+30} \lor \neg \left(z \leq 2 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m}{1} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (/ x_m 1.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m / 1.0);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (x_m / 1.0d0)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m / 1.0);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (x_m / 1.0)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(x_m / 1.0))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (x_m / 1.0);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(x$95$m / 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

   \[\frac{x \cdot \left(y + z\right)}{z} \]
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. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 95.6

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 95.6

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

4. Applied rewrites 95.6%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 54.5%

   \[\leadsto \frac{x}{\color{blue}{1}} \]
8. 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 2024313 
(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))