Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J

Percentage Accurate: 95.7% → 99.9%

Time: 11.7s

Alternatives: 13

Speedup: 0.8×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* (- 1.0 y) z))))

C

double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - 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 * (1.0d0 - ((1.0d0 - y) * z))
end function

Java

public static double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * z));
}

Python

def code(x, y, z):
	return x * (1.0 - ((1.0 - y) * z))

Julia

function code(x, y, z)
	return Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (1.0 - ((1.0 - y) * z));
end

Wolfram

code[x_, y_, z_] := N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* (- 1.0 y) z))))

C

double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - 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 * (1.0d0 - ((1.0d0 - y) * z))
end function

Java

public static double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * z));
}

Python

def code(x, y, z):
	return x * (1.0 - ((1.0 - y) * z))

Julia

function code(x, y, z)
	return Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (1.0 - ((1.0 - y) * z));
end

Wolfram

code[x_, y_, z_] := N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot \left(y + -1\right), z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - z \cdot \left(1 - y\right)\right)\\ \end{array} \end{array} \]

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-21)
    (fma (* x_m (+ y -1.0)) z x_m)
    (* x_m (- 1.0 (* z (- 1.0 y)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5e-21) {
		tmp = fma((x_m * (y + -1.0)), z, x_m);
	} else {
		tmp = x_m * (1.0 - (z * (1.0 - y)));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 5e-21)
		tmp = fma(Float64(x_m * Float64(y + -1.0)), z, x_m);
	else
		tmp = Float64(x_m * Float64(1.0 - Float64(z * Float64(1.0 - y))));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 5e-21], N[(N[(x$95$m * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] * z + x$95$m), $MachinePrecision], N[(x$95$m * N[(1.0 - N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999973e-21

   1. Initial program 94.7%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied rewrites 96.6%

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

   if 4.99999999999999973e-21 < x

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y + -1\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z \cdot \left(1 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := z \cdot \left(1 - y\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;z \cdot \left(x\_m \cdot y - x\_m\right)\\ \mathbf{elif}\;t\_0 \leq 1000:\\ \;\;\;\;x\_m - x\_m \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x\_m, -x\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (* z (- 1.0 y))))
   (*
    x_s
    (if (<= t_0 -100.0)
      (* z (- (* x_m y) x_m))
      (if (<= t_0 1000.0) (- x_m (* x_m z)) (* z (fma y 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 t_0 = z * (1.0 - y);
	double tmp;
	if (t_0 <= -100.0) {
		tmp = z * ((x_m * y) - x_m);
	} else if (t_0 <= 1000.0) {
		tmp = x_m - (x_m * z);
	} else {
		tmp = z * fma(y, 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)
	t_0 = Float64(z * Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(z * Float64(Float64(x_m * y) - x_m));
	elseif (t_0 <= 1000.0)
		tmp = Float64(x_m - Float64(x_m * z));
	else
		tmp = Float64(z * fma(y, x_m, Float64(-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_] := Block[{t$95$0 = N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -100.0], N[(z * N[(N[(x$95$m * y), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1000.0], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * x$95$m + (-x$95$m)), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) y) z) < -100

   1. Initial program 95.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x \cdot \left(z \cdot \left(y - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

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

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

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

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. lower--.f64 N/A

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

      26. lower-*.f64 95.8

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

   5. Applied rewrites 95.8%

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

   if -100 < (*.f64 (-.f64 #s(literal 1 binary64) y) z) < 1e3

   1. Initial program 100.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(1 - z\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 1e3 < (*.f64 (-.f64 #s(literal 1 binary64) y) z)

   1. Initial program 91.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x \cdot \left(z \cdot \left(y - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

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

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

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

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. lower--.f64 N/A

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

      26. lower-*.f64 91.9

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 91.9%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq -100:\\ \;\;\;\;z \cdot \left(x \cdot y - x\right)\\ \mathbf{elif}\;z \cdot \left(1 - y\right) \leq 1000:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, -x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := z \cdot \left(1 - y\right)\\ t_1 := z \cdot \left(x\_m \cdot y - x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1000:\\ \;\;\;\;x\_m - x\_m \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* z (- 1.0 y))) (t_1 (* z (- (* x_m y) x_m))))
   (*
    x_s
    (if (<= t_0 -100.0) t_1 (if (<= t_0 1000.0) (- x_m (* x_m z)) t_1)))))

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 = z * (1.0 - y);
	double t_1 = z * ((x_m * y) - x_m);
	double tmp;
	if (t_0 <= -100.0) {
		tmp = t_1;
	} else if (t_0 <= 1000.0) {
		tmp = x_m - (x_m * z);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = z * (1.0d0 - y)
    t_1 = z * ((x_m * y) - x_m)
    if (t_0 <= (-100.0d0)) then
        tmp = t_1
    else if (t_0 <= 1000.0d0) then
        tmp = x_m - (x_m * z)
    else
        tmp = t_1
    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 = z * (1.0 - y);
	double t_1 = z * ((x_m * y) - x_m);
	double tmp;
	if (t_0 <= -100.0) {
		tmp = t_1;
	} else if (t_0 <= 1000.0) {
		tmp = x_m - (x_m * z);
	} else {
		tmp = t_1;
	}
	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 = z * (1.0 - y)
	t_1 = z * ((x_m * y) - x_m)
	tmp = 0
	if t_0 <= -100.0:
		tmp = t_1
	elif t_0 <= 1000.0:
		tmp = x_m - (x_m * z)
	else:
		tmp = t_1
	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(z * Float64(1.0 - y))
	t_1 = Float64(z * Float64(Float64(x_m * y) - x_m))
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = t_1;
	elseif (t_0 <= 1000.0)
		tmp = Float64(x_m - Float64(x_m * z));
	else
		tmp = t_1;
	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 = z * (1.0 - y);
	t_1 = z * ((x_m * y) - x_m);
	tmp = 0.0;
	if (t_0 <= -100.0)
		tmp = t_1;
	elseif (t_0 <= 1000.0)
		tmp = x_m - (x_m * z);
	else
		tmp = t_1;
	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[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(N[(x$95$m * y), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -100.0], t$95$1, If[LessEqual[t$95$0, 1000.0], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) y) z) < -100 or 1e3 < (*.f64 (-.f64 #s(literal 1 binary64) y) z)

   1. Initial program 93.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x \cdot \left(z \cdot \left(y - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

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

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

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

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. lower--.f64 N/A

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

      26. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if -100 < (*.f64 (-.f64 #s(literal 1 binary64) y) z) < 1e3

   1. Initial program 100.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(1 - z\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq -100:\\ \;\;\;\;z \cdot \left(x \cdot y - x\right)\\ \mathbf{elif}\;z \cdot \left(1 - y\right) \leq 1000:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(y, x\_m, x\_m\right), z, x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \leq -0.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 2:\\ \;\;\;\;x\_m - x\_m \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (fma (fma y x_m x_m) z x_m)))
   (*
    x_s
    (if (<= (- 1.0 y) -0.4)
      t_0
      (if (<= (- 1.0 y) 2.0) (- x_m (* x_m z)) 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 = fma(fma(y, x_m, x_m), z, x_m);
	double tmp;
	if ((1.0 - y) <= -0.4) {
		tmp = t_0;
	} else if ((1.0 - y) <= 2.0) {
		tmp = x_m - (x_m * z);
	} 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 = fma(fma(y, x_m, x_m), z, x_m)
	tmp = 0.0
	if (Float64(1.0 - y) <= -0.4)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 2.0)
		tmp = Float64(x_m - Float64(x_m * z));
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(y * x$95$m + x$95$m), $MachinePrecision] * z + x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(1.0 - y), $MachinePrecision], -0.4], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 2.0], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -0.40000000000000002 or 2 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 91.5%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied rewrites 94.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. clear-num N/A

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

      12. flip-+ N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 92.2

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

      15. lift-+.f64 N/A

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

   6. Applied rewrites 94.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 94.4

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. neg-mul-1 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 92.3

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

   8. Applied rewrites 91.6%

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

   if -0.40000000000000002 < (-.f64 #s(literal 1 binary64) y) < 2

   1. Initial program 100.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(1 - z\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -0.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, x, x\right), z, x\right)\\ \mathbf{elif}\;1 - y \leq 2:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, x, x\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \leq -2 \cdot 10^{+77}:\\ \;\;\;\;y \cdot \left(x\_m \cdot z\right)\\ \mathbf{elif}\;1 - y \leq 2 \cdot 10^{+70}:\\ \;\;\;\;x\_m - x\_m \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x\_m \cdot y\right)\\ \end{array} \end{array} \]

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 (<= (- 1.0 y) -2e+77)
    (* y (* x_m z))
    (if (<= (- 1.0 y) 2e+70) (- x_m (* x_m z)) (* z (* x_m y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((1.0 - y) <= -2e+77) {
		tmp = y * (x_m * z);
	} else if ((1.0 - y) <= 2e+70) {
		tmp = x_m - (x_m * z);
	} else {
		tmp = z * (x_m * y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((1.0d0 - y) <= (-2d+77)) then
        tmp = y * (x_m * z)
    else if ((1.0d0 - y) <= 2d+70) then
        tmp = x_m - (x_m * z)
    else
        tmp = z * (x_m * y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((1.0 - y) <= -2e+77) {
		tmp = y * (x_m * z);
	} else if ((1.0 - y) <= 2e+70) {
		tmp = x_m - (x_m * z);
	} else {
		tmp = z * (x_m * y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (1.0 - y) <= -2e+77:
		tmp = y * (x_m * z)
	elif (1.0 - y) <= 2e+70:
		tmp = x_m - (x_m * z)
	else:
		tmp = z * (x_m * y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(1.0 - y) <= -2e+77)
		tmp = Float64(y * Float64(x_m * z));
	elseif (Float64(1.0 - y) <= 2e+70)
		tmp = Float64(x_m - Float64(x_m * z));
	else
		tmp = Float64(z * Float64(x_m * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((1.0 - y) <= -2e+77)
		tmp = y * (x_m * z);
	elseif ((1.0 - y) <= 2e+70)
		tmp = x_m - (x_m * z);
	else
		tmp = z * (x_m * y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(1.0 - y), $MachinePrecision], -2e+77], N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 - y), $MachinePrecision], 2e+70], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1.99999999999999997e77

   1. Initial program 88.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 72.7

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

   5. Applied rewrites 72.7%

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

   7. Applied rewrites 76.4%

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

   if -1.99999999999999997e77 < (-.f64 #s(literal 1 binary64) y) < 2.00000000000000015e70

   1. Initial program 99.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(1 - z\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if 2.00000000000000015e70 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 91.3%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -2 \cdot 10^{+77}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;1 - y \leq 2 \cdot 10^{+70}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := z \cdot \left(x\_m \cdot y\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \leq -2 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 2 \cdot 10^{+70}:\\ \;\;\;\;x\_m - x\_m \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* z (* x_m y))))
   (*
    x_s
    (if (<= (- 1.0 y) -2e+77)
      t_0
      (if (<= (- 1.0 y) 2e+70) (- x_m (* x_m z)) 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 = z * (x_m * y);
	double tmp;
	if ((1.0 - y) <= -2e+77) {
		tmp = t_0;
	} else if ((1.0 - y) <= 2e+70) {
		tmp = x_m - (x_m * z);
	} 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 = z * (x_m * y)
    if ((1.0d0 - y) <= (-2d+77)) then
        tmp = t_0
    else if ((1.0d0 - y) <= 2d+70) then
        tmp = x_m - (x_m * z)
    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 = z * (x_m * y);
	double tmp;
	if ((1.0 - y) <= -2e+77) {
		tmp = t_0;
	} else if ((1.0 - y) <= 2e+70) {
		tmp = x_m - (x_m * z);
	} 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 = z * (x_m * y)
	tmp = 0
	if (1.0 - y) <= -2e+77:
		tmp = t_0
	elif (1.0 - y) <= 2e+70:
		tmp = x_m - (x_m * z)
	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(z * Float64(x_m * y))
	tmp = 0.0
	if (Float64(1.0 - y) <= -2e+77)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 2e+70)
		tmp = Float64(x_m - Float64(x_m * z));
	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 = z * (x_m * y);
	tmp = 0.0;
	if ((1.0 - y) <= -2e+77)
		tmp = t_0;
	elseif ((1.0 - y) <= 2e+70)
		tmp = x_m - (x_m * z);
	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[(z * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(1.0 - y), $MachinePrecision], -2e+77], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 2e+70], N[(x$95$m - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1.99999999999999997e77 or 2.00000000000000015e70 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 89.8%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

   if -1.99999999999999997e77 < (-.f64 #s(literal 1 binary64) y) < 2.00000000000000015e70

   1. Initial program 99.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(1 - z\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -2 \cdot 10^{+77}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;1 - y \leq 2 \cdot 10^{+70}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot \left(y + -1\right), z, x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-26}:\\ \;\;\;\;x\_m + x\_m \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (fma (* x_m (+ y -1.0)) z x_m)))
   (*
    x_s
    (if (<= z -1.75e-6) t_0 (if (<= z 2e-26) (+ x_m (* x_m (* y z))) 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 = fma((x_m * (y + -1.0)), z, x_m);
	double tmp;
	if (z <= -1.75e-6) {
		tmp = t_0;
	} else if (z <= 2e-26) {
		tmp = x_m + (x_m * (y * z));
	} 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 = fma(Float64(x_m * Float64(y + -1.0)), z, x_m)
	tmp = 0.0
	if (z <= -1.75e-6)
		tmp = t_0;
	elseif (z <= 2e-26)
		tmp = Float64(x_m + Float64(x_m * Float64(y * z)));
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(x$95$m * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] * z + x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.75e-6], t$95$0, If[LessEqual[z, 2e-26], N[(x$95$m + N[(x$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.74999999999999997e-6 or 2.0000000000000001e-26 < z

   1. Initial program 92.1%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied rewrites 99.9%

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

   if -1.74999999999999997e-6 < z < 2.0000000000000001e-26

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y + -1\right), z, x\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-26}:\\ \;\;\;\;x + x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y + -1\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.93:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x\_m, -x\_m\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\_m + x\_m \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x\_m \cdot y - x\_m\right)\\ \end{array} \end{array} \]

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 (<= z -0.93)
    (* z (fma y x_m (- x_m)))
    (if (<= z 1.0) (+ x_m (* x_m (* y z))) (* z (- (* x_m 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) {
	double tmp;
	if (z <= -0.93) {
		tmp = z * fma(y, x_m, -x_m);
	} else if (z <= 1.0) {
		tmp = x_m + (x_m * (y * z));
	} else {
		tmp = z * ((x_m * y) - 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 (z <= -0.93)
		tmp = Float64(z * fma(y, x_m, Float64(-x_m)));
	elseif (z <= 1.0)
		tmp = Float64(x_m + Float64(x_m * Float64(y * z)));
	else
		tmp = Float64(z * Float64(Float64(x_m * y) - 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[z, -0.93], N[(z * N[(y * x$95$m + (-x$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(x$95$m + N[(x$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(x$95$m * y), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -0.930000000000000049

   1. Initial program 93.2%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x \cdot \left(z \cdot \left(y - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

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

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

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

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. lower--.f64 N/A

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

      26. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 98.3%

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

   if -0.930000000000000049 < z < 1

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if 1 < z

   1. Initial program 90.5%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x \cdot \left(z \cdot \left(y - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

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

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

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

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. lower--.f64 N/A

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

      26. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x - x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.93:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, -x\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.25 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot \left(y + -1\right), z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + -1, x\_m \cdot z, x\_m\right)\\ \end{array} \end{array} \]

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 1.25e-82)
    (fma (* x_m (+ y -1.0)) z x_m)
    (fma (+ y -1.0) (* x_m z) 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 <= 1.25e-82) {
		tmp = fma((x_m * (y + -1.0)), z, x_m);
	} else {
		tmp = fma((y + -1.0), (x_m * z), 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 <= 1.25e-82)
		tmp = fma(Float64(x_m * Float64(y + -1.0)), z, x_m);
	else
		tmp = fma(Float64(y + -1.0), Float64(x_m * z), 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, 1.25e-82], N[(N[(x$95$m * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] * z + x$95$m), $MachinePrecision], N[(N[(y + -1.0), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision] + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.25e-82

   1. Initial program 94.7%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied rewrites 96.3%

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

   if 1.25e-82 < x

   1. Initial program 98.8%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y + -1, z \cdot x, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y + -1\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + -1, x \cdot z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := z \cdot \left(-x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 91.7%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x \cdot \left(z \cdot \left(y - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

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

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

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

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. lower--.f64 N/A

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

      26. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto z \cdot \left(-1 \cdot \color{blue}{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 58.7%

      \[\leadsto z \cdot \left(-x\right) \]

   if -1 < z < 1

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 70.7%

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

Alternative 11: 65.6% accurate, 1.9× speedup

Math

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

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 (* 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) {
	return x_s * (x_m - (x_m * 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 - (x_m * 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 - (x_m * 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 - (x_m * 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(x_m * 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 - (x_m * 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[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x \cdot \left(1 - z\right)} \]
4. Step-by-step derivation

   1. distribute-lft-out-- N/A

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

   2. *-rgt-identity N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 66.6

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

5. Applied rewrites 66.6%

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

6. Final simplification 66.6%

   \[\leadsto x - x \cdot z \]
7. Add Preprocessing

Alternative 12: 37.7% accurate, 2.8× speedup

Math

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

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 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 95.9%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

5. Applied rewrites 37.8%

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

Alternative 13: 6.0% accurate, 2.8× speedup

Math

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

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 (* z 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 * (z * 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 * (z * 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 * (z * 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 * (z * 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(z * 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 * (z * 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[(z * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x \cdot \left(z \cdot \left(y - 1\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. cancel-sign-sub N/A

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

   8. mul-1-neg N/A

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

   9. *-rgt-identity N/A

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

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

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

   11. mul-1-neg N/A

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

   12. *-commutative N/A

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

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

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

   14. associate-*r* N/A

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

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

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

   16. distribute-rgt-neg-in N/A

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

   17. lower-*.f64 N/A

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

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

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

   19. *-lft-identity N/A

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

   20. unsub-neg N/A

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

   21. +-commutative N/A

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

   22. distribute-neg-in N/A

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

   23. remove-double-neg N/A

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

   24. unsub-neg N/A

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

   25. lower--.f64 N/A

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

   26. lower-*.f64 60.5

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

5. Applied rewrites 60.5%

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

6. Taylor expanded in y around 0

   \[\leadsto z \cdot \left(-1 \cdot \color{blue}{x}\right) \]
7. Step-by-step derivation

8. Applied rewrites 30.9%

   \[\leadsto z \cdot \left(-x\right) \]
9. Step-by-step derivation

10. Applied rewrites 5.1%

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

11. Final simplification 5.1%

    \[\leadsto z \cdot x \]
12. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ t_1 := x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{if}\;t\_0 < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* (- 1.0 y) z))))
        (t_1 (+ x (* (- 1.0 y) (* (- z) x)))))
   (if (< t_0 -1.618195973607049e+50)
     t_1
     (if (< t_0 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = x * (1.0 - ((1.0 - y) * z));
	double t_1 = x + ((1.0 - y) * (-z * x));
	double tmp;
	if (t_0 < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (t_0 < 3.892237649663903e+134) {
		tmp = ((x * y) * z) - ((x * z) - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (1.0d0 - ((1.0d0 - y) * z))
    t_1 = x + ((1.0d0 - y) * (-z * x))
    if (t_0 < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (t_0 < 3.892237649663903d+134) then
        tmp = ((x * y) * z) - ((x * z) - x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (1.0 - ((1.0 - y) * z));
	double t_1 = x + ((1.0 - y) * (-z * x));
	double tmp;
	if (t_0 < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (t_0 < 3.892237649663903e+134) {
		tmp = ((x * y) * z) - ((x * z) - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (1.0 - ((1.0 - y) * z))
	t_1 = x + ((1.0 - y) * (-z * x))
	tmp = 0
	if t_0 < -1.618195973607049e+50:
		tmp = t_1
	elif t_0 < 3.892237649663903e+134:
		tmp = ((x * y) * z) - ((x * z) - x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
	t_1 = Float64(x + Float64(Float64(1.0 - y) * Float64(Float64(-z) * x)))
	tmp = 0.0
	if (t_0 < -1.618195973607049e+50)
		tmp = t_1;
	elseif (t_0 < 3.892237649663903e+134)
		tmp = Float64(Float64(Float64(x * y) * z) - Float64(Float64(x * z) - x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (1.0 - ((1.0 - y) * z));
	t_1 = x + ((1.0 - y) * (-z * x));
	tmp = 0.0;
	if (t_0 < -1.618195973607049e+50)
		tmp = t_1;
	elseif (t_0 < 3.892237649663903e+134)
		tmp = ((x * y) * z) - ((x * z) - x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(N[(1.0 - y), $MachinePrecision] * N[((-z) * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$0, -1.618195973607049e+50], t$95$1, If[Less[t$95$0, 3.892237649663903e+134], N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] - N[(N[(x * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024238 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* x (- 1 (* (- 1 y) z))) -161819597360704900000000000000000000000000000000000) (+ x (* (- 1 y) (* (- z) x))) (if (< (* x (- 1 (* (- 1 y) z))) 389223764966390300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1 y) (* (- z) x))))))

  (* x (- 1.0 (* (- 1.0 y) z))))