Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.3% → 98.0%

Time: 6.8s

Alternatives: 5

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: 84.3% 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: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{z}, y, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\_m + x\_m\\ \end{array} \end{array} \]

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 5e-31) (fma (/ x_m z) y x_m) (+ (* (/ 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 <= 5e-31) {
		tmp = fma((x_m / z), y, x_m);
	} else {
		tmp = ((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 (x_m <= 5e-31)
		tmp = fma(Float64(x_m / z), y, x_m);
	else
		tmp = Float64(Float64(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[x$95$m, 5e-31], N[(N[(x$95$m / z), $MachinePrecision] * y + x$95$m), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] * x$95$m), $MachinePrecision] + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e-31

   1. Initial program 88.5%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 55.1

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

   4. Applied rewrites 55.1%

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

   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. div-add N/A

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

      3. distribute-rgt-in N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

         \[\leadsto \frac{y \cdot \color{blue}{1}}{z} \cdot x + \frac{{z}^{2}}{{z}^{2}} \cdot x \]

      8. *-rgt-identity N/A

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

      9. associate-*l/ N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. *-inverses N/A

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

      13. *-lft-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 95.4

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

   7. Applied rewrites 95.4%

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

   if 5e-31 < x

   1. Initial program 76.5%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

      9. *-inverses N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x + x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \left(y + z\right)}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (+ y z)) z)))
   (*
    x_s
    (if (or (<= t_0 0.0) (not (<= t_0 5e+119)))
      (* (/ x_m z) y)
      (/ (* z x_m) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (y + z)) / z;
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+119)) {
		tmp = (x_m / z) * y;
	} else {
		tmp = (z * x_m) / z;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (y + z)) / z
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+119))) then
        tmp = (x_m / z) * y
    else
        tmp = (z * x_m) / z
    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 t_0 = (x_m * (y + z)) / z;
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+119)) {
		tmp = (x_m / z) * y;
	} else {
		tmp = (z * x_m) / z;
	}
	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):
	t_0 = (x_m * (y + z)) / z
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+119):
		tmp = (x_m / z) * y
	else:
		tmp = (z * x_m) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(y + z)) / z)
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+119))
		tmp = Float64(Float64(x_m / z) * y);
	else
		tmp = Float64(Float64(z * x_m) / z);
	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)
	t_0 = (x_m * (y + z)) / z;
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+119)))
		tmp = (x_m / z) * y;
	else
		tmp = (z * x_m) / z;
	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_] := Block[{t$95$0 = N[(N[(x$95$m * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+119]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(z * x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < 0.0 or 4.9999999999999999e119 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 80.1%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. sqrt-prod N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. distribute-lft-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 59.4

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

   7. Applied rewrites 59.4%

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

   if 0.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < 4.9999999999999999e119

   1. Initial program 99.7%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 69.3

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

   5. Applied rewrites 69.3%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq 0 \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \leq 5 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{z}, y, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x\_m, x\_m\right)\\ \end{array} \end{array} \]

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 5e-31) (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 <= 5e-31) {
		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 (x_m <= 5e-31)
		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[x$95$m, 5e-31], 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 x < 5e-31

   1. Initial program 88.5%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 55.1

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

   4. Applied rewrites 55.1%

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

   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. div-add N/A

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

      3. distribute-rgt-in N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

         \[\leadsto \frac{y \cdot \color{blue}{1}}{z} \cdot x + \frac{{z}^{2}}{{z}^{2}} \cdot x \]

      8. *-rgt-identity N/A

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

      9. associate-*l/ N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. *-inverses N/A

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

      13. *-lft-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 95.4

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

   7. Applied rewrites 95.4%

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

   if 5e-31 < x

   1. Initial program 76.5%

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

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 1.1× speedup

Math

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

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 (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) {
	return x_s * fma((y / z), x_m, x_m);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * fma(Float64(y / z), x_m, x_m))
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[(N[(y / z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.5%

   \[\frac{x \cdot \left(y + z\right)}{z} \]
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 95.5

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

5. Applied rewrites 95.5%

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

Alternative 5: 48.2% accurate, 1.2× speedup

Math

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

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 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) {
	return x_s * ((x_m / z) * y);
}

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 / z) * y)
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 / z) * y);
}

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 / z) * y)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(Float64(x_m / z) * y))
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 / z) * y);
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[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

   1. lift-+.f64 N/A

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

   2. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow1 N/A

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

   5. pow2 N/A

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

   6. sqrt-prod N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 44.6

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

4. Applied rewrites 44.6%

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

5. Taylor expanded in y around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. rem-square-sqrt N/A

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

   7. distribute-lft-neg-in N/A

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

   8. mul-1-neg N/A

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

   9. remove-double-neg N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 51.5

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

7. Applied rewrites 51.5%

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

8. Final simplification 51.5%

   \[\leadsto \frac{x}{z} \cdot y \]
9. Add Preprocessing

Developer Target 1: 96.4% 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 2024320 
(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))