Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.4% → 97.7%

Time: 6.3s

Alternatives: 6

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: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{z}, y, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x\_m, x\_m\right)\\ \end{array} \end{array} \]

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-61) (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-61) {
		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-61)
		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-61], 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 < 4.9999999999999999e-61

   1. Initial program 84.0%

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

   3. Taylor expanded in z around 0

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

      1. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. associate-*r/ N/A

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

      10. unsub-neg N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. distribute-lft-out N/A

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

      19. associate-/l* N/A

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

      20. associate-*l/ N/A

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

      21. *-rgt-identity N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if 4.9999999999999999e-61 < x

   1. Initial program 72.6%

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

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

      4. associate-/l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. flip-+ N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower--.f64 N/A

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

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

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower--.f64 42.7

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

   4. Applied rewrites 42.7%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

Alternative 2: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+34}:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x\_m}{z}\\ \end{array} \end{array} \]

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 (<= y -4.8e-109)
    (* (/ x_m z) y)
    (if (<= y 3.6e+34) (* 1.0 x_m) (/ (* y 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 tmp;
	if (y <= -4.8e-109) {
		tmp = (x_m / z) * y;
	} else if (y <= 3.6e+34) {
		tmp = 1.0 * x_m;
	} else {
		tmp = (y * 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) :: tmp
    if (y <= (-4.8d-109)) then
        tmp = (x_m / z) * y
    else if (y <= 3.6d+34) then
        tmp = 1.0d0 * x_m
    else
        tmp = (y * 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 tmp;
	if (y <= -4.8e-109) {
		tmp = (x_m / z) * y;
	} else if (y <= 3.6e+34) {
		tmp = 1.0 * x_m;
	} else {
		tmp = (y * 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):
	tmp = 0
	if y <= -4.8e-109:
		tmp = (x_m / z) * y
	elif y <= 3.6e+34:
		tmp = 1.0 * x_m
	else:
		tmp = (y * 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)
	tmp = 0.0
	if (y <= -4.8e-109)
		tmp = Float64(Float64(x_m / z) * y);
	elseif (y <= 3.6e+34)
		tmp = Float64(1.0 * x_m);
	else
		tmp = Float64(Float64(y * 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)
	tmp = 0.0;
	if (y <= -4.8e-109)
		tmp = (x_m / z) * y;
	elseif (y <= 3.6e+34)
		tmp = 1.0 * x_m;
	else
		tmp = (y * 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_] := N[(x$95$s * If[LessEqual[y, -4.8e-109], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 3.6e+34], N[(1.0 * x$95$m), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -4.79999999999999977e-109

   1. Initial program 84.6%

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

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

      4. associate-/l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. flip-+ N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower--.f64 N/A

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

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

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower--.f64 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 93.5

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   9. 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 73.0

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

   10. Applied rewrites 73.0%

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

   if -4.79999999999999977e-109 < y < 3.6e34

   1. Initial program 74.2%

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 81.3%

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

   if 3.6e34 < y

   1. Initial program 90.3%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

Alternative 3: 71.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m}{z} \cdot y\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+34}:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 z) y)))
   (* x_s (if (<= y -4.8e-109) t_0 (if (<= y 3.6e+34) (* 1.0 x_m) t_0)))))

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 / z) * y;
	double tmp;
	if (y <= -4.8e-109) {
		tmp = t_0;
	} else if (y <= 3.6e+34) {
		tmp = 1.0 * x_m;
	} else {
		tmp = t_0;
	}
	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 / z) * y
    if (y <= (-4.8d-109)) then
        tmp = t_0
    else if (y <= 3.6d+34) then
        tmp = 1.0d0 * x_m
    else
        tmp = t_0
    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 / z) * y;
	double tmp;
	if (y <= -4.8e-109) {
		tmp = t_0;
	} else if (y <= 3.6e+34) {
		tmp = 1.0 * x_m;
	} else {
		tmp = t_0;
	}
	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 / z) * y
	tmp = 0
	if y <= -4.8e-109:
		tmp = t_0
	elif y <= 3.6e+34:
		tmp = 1.0 * x_m
	else:
		tmp = t_0
	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 / z) * y)
	tmp = 0.0
	if (y <= -4.8e-109)
		tmp = t_0;
	elseif (y <= 3.6e+34)
		tmp = Float64(1.0 * x_m);
	else
		tmp = t_0;
	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 / z) * y;
	tmp = 0.0;
	if (y <= -4.8e-109)
		tmp = t_0;
	elseif (y <= 3.6e+34)
		tmp = 1.0 * x_m;
	else
		tmp = t_0;
	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 / z), $MachinePrecision] * y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -4.8e-109], t$95$0, If[LessEqual[y, 3.6e+34], N[(1.0 * x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.79999999999999977e-109 or 3.6e34 < y

   1. Initial program 87.2%

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

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

      4. associate-/l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. flip-+ N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower--.f64 N/A

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

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

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower--.f64 44.7

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

   4. Applied rewrites 44.7%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 92.1

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

   7. Applied rewrites 92.1%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   9. 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 72.5

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

   10. Applied rewrites 72.5%

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

   if -4.79999999999999977e-109 < y < 3.6e34

   1. Initial program 74.2%

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 81.3%

      \[\leadsto \color{blue}{1} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.1% accurate, 0.8× speedup

Math

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

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 z)))))

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 + z)));
}

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 + z)))
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 + z)));
}

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 + z)))

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 / Float64(z / Float64(y + z))))
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 + z)));
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 / N[(z / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

   \[\frac{x \cdot \left(y + z\right)}{z} \]
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 96.2

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

   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 96.2

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

4. Applied rewrites 96.2%

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

5. Final simplification 96.2%

   \[\leadsto \frac{x}{\frac{z}{y + z}} \]
6. Add Preprocessing

Alternative 5: 94.1% accurate, 1.1× speedup

Math

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

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 (/ x_m z) y 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((x_m / z), y, 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(x_m / z), y, 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[(x$95$m / z), $MachinePrecision] * y + x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in z around 0

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

   1. distribute-lft-in N/A

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

   2. associate-/l* N/A

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

   3. +-commutative N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

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

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

   7. div-sub N/A

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

   8. *-inverses N/A

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

   9. associate-*r/ N/A

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

   10. unsub-neg N/A

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

   11. mul-1-neg N/A

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

   12. remove-double-neg N/A

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

   13. +-commutative N/A

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

   14. metadata-eval N/A

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

   15. sub-neg N/A

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

   16. sub-neg N/A

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

   17. metadata-eval N/A

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

   18. distribute-lft-out N/A

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

   19. associate-/l* N/A

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

   20. associate-*l/ N/A

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

   21. *-rgt-identity N/A

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

   22. lower-fma.f64 N/A

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

   23. lower-/.f64 93.6

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

5. Applied rewrites 93.6%

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

Alternative 6: 49.8% accurate, 3.3× speedup

Math

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

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 (* 1.0 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 * (1.0 * x_m);
}

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 * (1.0d0 * x_m)
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 * (1.0 * x_m);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (1.0 * 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 * Float64(1.0 * x_m))
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 * (1.0 * 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[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 95.8

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 95.8

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

4. Applied rewrites 95.8%

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

5. Taylor expanded in z around inf

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

7. Applied rewrites 51.4%

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

Developer Target 1: 96.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x / (z / (y + z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024270 
(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))