Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.4% → 98.1%

Time: 5.1s

Alternatives: 4

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: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{z}, y, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x\_m, x\_m\right)\\ \end{array} \end{array} \]

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 6e-23) (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 <= 6e-23) {
		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 <= 6e-23)
		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, 6e-23], 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 < 6.00000000000000006e-23

   1. Initial program 88.9%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 94.8

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

      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 94.8

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

   4. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{z + y}}} \]
   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 95.3

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

   6. Applied rewrites 95.3%

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

   if 6.00000000000000006e-23 < x

   1. Initial program 80.5%

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

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

      2. *-commutative 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. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. associate-*r/ N/A

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

      10. unsub-neg N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 N/A

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

      16. 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. Add Preprocessing

Alternative 2: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := y \cdot \frac{x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-130}:\\ \;\;\;\;\frac{x\_m}{1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* y (/ x_m z))))
   (* x_s (if (<= y -1.8e+49) t_0 (if (<= y 4.3e-130) (/ x_m 1.0) 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 = y * (x_m / z);
	double tmp;
	if (y <= -1.8e+49) {
		tmp = t_0;
	} else if (y <= 4.3e-130) {
		tmp = x_m / 1.0;
	} 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 = y * (x_m / z)
    if (y <= (-1.8d+49)) then
        tmp = t_0
    else if (y <= 4.3d-130) then
        tmp = x_m / 1.0d0
    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 = y * (x_m / z);
	double tmp;
	if (y <= -1.8e+49) {
		tmp = t_0;
	} else if (y <= 4.3e-130) {
		tmp = x_m / 1.0;
	} 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 = y * (x_m / z)
	tmp = 0
	if y <= -1.8e+49:
		tmp = t_0
	elif y <= 4.3e-130:
		tmp = x_m / 1.0
	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(y * Float64(x_m / z))
	tmp = 0.0
	if (y <= -1.8e+49)
		tmp = t_0;
	elseif (y <= 4.3e-130)
		tmp = Float64(x_m / 1.0);
	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 = y * (x_m / z);
	tmp = 0.0;
	if (y <= -1.8e+49)
		tmp = t_0;
	elseif (y <= 4.3e-130)
		tmp = x_m / 1.0;
	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[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.8e+49], t$95$0, If[LessEqual[y, 4.3e-130], N[(x$95$m / 1.0), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.79999999999999998e49 or 4.30000000000000029e-130 < y

   1. Initial program 87.7%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 93.5

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

      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 93.5

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

   4. Applied rewrites 93.5%

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

   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 76.0

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

   7. Applied rewrites 76.0%

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

   if -1.79999999999999998e49 < y < 4.30000000000000029e-130

   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 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 80.4%

      \[\leadsto \frac{x}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-130}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% 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 86.4%

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

3. Taylor expanded in x around 0

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

   1. associate-/l* N/A

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

   2. *-commutative 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. *-lft-identity N/A

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

   5. metadata-eval N/A

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

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

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

   7. div-sub N/A

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

   8. *-inverses N/A

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

   9. associate-*r/ N/A

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

   10. unsub-neg N/A

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

   11. mul-1-neg N/A

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

   12. remove-double-neg N/A

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

   13. +-commutative N/A

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

   14. distribute-lft1-in N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-/.f64 96.3

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

5. Applied rewrites 96.3%

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

Alternative 4: 51.0% 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 86.4%

   \[\frac{x \cdot \left(y + z\right)}{z} \]
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 96.3

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

   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 96.3

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

4. Applied rewrites 96.3%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 48.0%

   \[\leadsto \frac{x}{\color{blue}{1}} \]
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 2024332 
(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))