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

Percentage Accurate: 95.8% → 99.7%

Time: 7.9s

Alternatives: 8

Speedup: 1.1×

• 20.9% 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: 95.8% 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: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.8 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(\left(-1 + y\right) \cdot x\_m, z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-1 + y\right) \cdot z, x\_m, x\_m\right)\\ \end{array} \end{array} \]

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 5.8e-66)
    (fma (* (+ -1.0 y) x_m) z x_m)
    (fma (* (+ -1.0 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 <= 5.8e-66) {
		tmp = fma(((-1.0 + y) * x_m), z, x_m);
	} else {
		tmp = fma(((-1.0 + 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 <= 5.8e-66)
		tmp = fma(Float64(Float64(-1.0 + y) * x_m), z, x_m);
	else
		tmp = fma(Float64(Float64(-1.0 + 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, 5.8e-66], N[(N[(N[(-1.0 + y), $MachinePrecision] * x$95$m), $MachinePrecision] * z + x$95$m), $MachinePrecision], N[(N[(N[(-1.0 + y), $MachinePrecision] * z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.80000000000000023e-66

   1. Initial program 93.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 93.9

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

   5. Applied rewrites 93.9%

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

   7. Applied rewrites 98.8%

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

   if 5.80000000000000023e-66 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 2: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \leq -500 \lor \neg \left(1 - y \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot x\_m, z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x\_m, x\_m\right)\\ \end{array} \end{array} \]

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 (or (<= (- 1.0 y) -500.0) (not (<= (- 1.0 y) 2.0)))
    (fma (* y x_m) z x_m)
    (fma (- 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 (((1.0 - y) <= -500.0) || !((1.0 - y) <= 2.0)) {
		tmp = fma((y * x_m), z, x_m);
	} else {
		tmp = fma(-z, x_m, x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((Float64(1.0 - y) <= -500.0) || !(Float64(1.0 - y) <= 2.0))
		tmp = fma(Float64(y * x_m), z, x_m);
	else
		tmp = fma(Float64(-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[Or[LessEqual[N[(1.0 - y), $MachinePrecision], -500.0], N[Not[LessEqual[N[(1.0 - y), $MachinePrecision], 2.0]], $MachinePrecision]], N[(N[(y * x$95$m), $MachinePrecision] * z + x$95$m), $MachinePrecision], N[((-z) * x$95$m + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -500 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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 91.5

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 96.1%

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

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot y, z, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.5%

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

   if -500 < (-.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 x around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -500 \lor \neg \left(1 - y \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \leq -500:\\ \;\;\;\;\left(x\_m \cdot y\right) \cdot z + x\_m\\ \mathbf{elif}\;1 - y \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-z, x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x\_m, z, x\_m\right)\\ \end{array} \end{array} \]

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 (<= (- 1.0 y) -500.0)
    (+ (* (* x_m y) z) x_m)
    (if (<= (- 1.0 y) 2.0) (fma (- z) x_m x_m) (fma (* y 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 ((1.0 - y) <= -500.0) {
		tmp = ((x_m * y) * z) + x_m;
	} else if ((1.0 - y) <= 2.0) {
		tmp = fma(-z, x_m, x_m);
	} else {
		tmp = fma((y * 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 (Float64(1.0 - y) <= -500.0)
		tmp = Float64(Float64(Float64(x_m * y) * z) + x_m);
	elseif (Float64(1.0 - y) <= 2.0)
		tmp = fma(Float64(-z), x_m, x_m);
	else
		tmp = fma(Float64(y * 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[N[(1.0 - y), $MachinePrecision], -500.0], N[(N[(N[(x$95$m * y), $MachinePrecision] * z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[N[(1.0 - y), $MachinePrecision], 2.0], N[((-z) * x$95$m + x$95$m), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] * z + x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -500

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 93.1

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

   5. Applied rewrites 93.1%

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

   7. Applied rewrites 94.4%

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

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot y, z, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 93.4%

       \[\leadsto \mathsf{fma}\left(y \cdot x, z, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 93.4%

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

   if -500 < (-.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 x around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.6%

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

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

   1. Initial program 89.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 89.5

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

   5. Applied rewrites 89.5%

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot y, z, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 98.2%

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

Alternative 4: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \leq -2 \cdot 10^{+42} \lor \neg \left(1 - y \leq 2 \cdot 10^{+39}\right):\\ \;\;\;\;\left(y \cdot x\_m\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x\_m, x\_m\right)\\ \end{array} \end{array} \]

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 (or (<= (- 1.0 y) -2e+42) (not (<= (- 1.0 y) 2e+39)))
    (* (* y x_m) z)
    (fma (- 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 (((1.0 - y) <= -2e+42) || !((1.0 - y) <= 2e+39)) {
		tmp = (y * x_m) * z;
	} else {
		tmp = fma(-z, x_m, x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((Float64(1.0 - y) <= -2e+42) || !(Float64(1.0 - y) <= 2e+39))
		tmp = Float64(Float64(y * x_m) * z);
	else
		tmp = fma(Float64(-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[Or[LessEqual[N[(1.0 - y), $MachinePrecision], -2e+42], N[Not[LessEqual[N[(1.0 - y), $MachinePrecision], 2e+39]], $MachinePrecision]], N[(N[(y * x$95$m), $MachinePrecision] * z), $MachinePrecision], N[((-z) * x$95$m + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -2.00000000000000009e42 or 1.99999999999999988e39 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 90.5%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

   7. Applied rewrites 76.7%

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

   if -2.00000000000000009e42 < (-.f64 #s(literal 1 binary64) y) < 1.99999999999999988e39

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.6%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -2 \cdot 10^{+42} \lor \neg \left(1 - y \leq 2 \cdot 10^{+39}\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 9.2 \cdot 10^{-23}\right):\\ \;\;\;\;x\_m \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (or (<= z -1.0) (not (<= z 9.2e-23))) (* x_m (- z)) (* 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) {
	double tmp;
	if ((z <= -1.0) || !(z <= 9.2e-23)) {
		tmp = x_m * -z;
	} else {
		tmp = x_m * 1.0;
	}
	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) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 9.2d-23))) then
        tmp = x_m * -z
    else
        tmp = x_m * 1.0d0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 9.2e-23)) {
		tmp = x_m * -z;
	} else {
		tmp = x_m * 1.0;
	}
	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):
	tmp = 0
	if (z <= -1.0) or not (z <= 9.2e-23):
		tmp = x_m * -z
	else:
		tmp = x_m * 1.0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 9.2e-23))
		tmp = Float64(x_m * Float64(-z));
	else
		tmp = Float64(x_m * 1.0);
	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)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 9.2e-23)))
		tmp = x_m * -z;
	else
		tmp = x_m * 1.0;
	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_] := N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 9.2e-23]], $MachinePrecision]], N[(x$95$m * (-z)), $MachinePrecision], N[(x$95$m * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 9.2000000000000004e-23 < z

   1. Initial program 91.8%

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

   3. Taylor expanded in z around inf

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

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

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-rgt-in N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

      19. fp-cancel-sign-sub-inv N/A

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

      20. distribute-neg-in N/A

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

      21. metadata-eval N/A

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

      22. mul-1-neg N/A

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

      23. remove-double-neg N/A

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

      24. lower-+.f64 91.3

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.3%

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

   if -1 < z < 9.2000000000000004e-23

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 75.6%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 9.2 \cdot 10^{-23}\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.8% accurate, 1.1× speedup

Math

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

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 (* (+ -1.0 y) 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) {
	return x_s * fma(((-1.0 + y) * x_m), 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 * fma(Float64(Float64(-1.0 + y) * x_m), 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[(N[(N[(-1.0 + y), $MachinePrecision] * x$95$m), $MachinePrecision] * 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 x around 0

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

   1. *-commutative N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. lower-fma.f64 N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. lower-*.f64 N/A

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

   11. mul-1-neg N/A

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

   12. *-lft-identity N/A

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

   13. metadata-eval N/A

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

   14. fp-cancel-sign-sub-inv N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. mul-1-neg N/A

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

   18. remove-double-neg N/A

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

   19. lower-+.f64 95.9

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

5. Applied rewrites 95.9%

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

7. Applied rewrites 98.1%

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

Alternative 7: 65.3% accurate, 1.9× speedup

Math

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

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 (- 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(-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(-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[((-z) * x$95$m + 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 x around 0

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

   1. *-commutative N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. lower-fma.f64 N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. lower-*.f64 N/A

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

   11. mul-1-neg N/A

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

   12. *-lft-identity N/A

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

   13. metadata-eval N/A

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

   14. fp-cancel-sign-sub-inv N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. mul-1-neg N/A

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

   18. remove-double-neg N/A

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

   19. lower-+.f64 95.9

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

5. Applied rewrites 95.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 66.3%

   \[\leadsto \mathsf{fma}\left(-z, x, x\right) \]
9. Add Preprocessing

Alternative 8: 38.4% 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 y around 0

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

   1. lower--.f64 66.3

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

5. Applied rewrites 66.3%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 39.7%

   \[\leadsto x \cdot \color{blue}{1} \]
9. 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 2024326 
(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))))