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

Percentage Accurate: 95.8% → 99.4%

Time: 8.2s

Alternatives: 11

Speedup: 0.8×

• 20.6% 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: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot x\_m, z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - 1, z \cdot x\_m, x\_m\right)\\ \end{array} \end{array} \]

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 1e+80)
    (fma (* (- y 1.0) x_m) z x_m)
    (fma (- y 1.0) (* 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 <= 1e+80) {
		tmp = fma(((y - 1.0) * x_m), z, x_m);
	} else {
		tmp = fma((y - 1.0), (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 <= 1e+80)
		tmp = fma(Float64(Float64(y - 1.0) * x_m), z, x_m);
	else
		tmp = fma(Float64(y - 1.0), 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[LessEqual[x$95$m, 1e+80], N[(N[(N[(y - 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * z + x$95$m), $MachinePrecision], N[(N[(y - 1.0), $MachinePrecision] * N[(z * x$95$m), $MachinePrecision] + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e80

   1. Initial program 95.2%

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

   4. Applied rewrites 96.7%

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

   if 1e80 < x

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - 1, z \cdot x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -200000:\\ \;\;\;\;z \cdot \left(\left(y - 1\right) \cdot x\_m\right)\\ \mathbf{elif}\;t\_0 \leq 20000000000:\\ \;\;\;\;\mathsf{fma}\left(-x\_m, z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\_m\right) \cdot \left(y - 1\right)\\ \end{array} \end{array} \end{array} \]

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 (* (- 1.0 y) z)))
   (*
    x_s
    (if (<= t_0 -200000.0)
      (* z (* (- y 1.0) x_m))
      (if (<= t_0 20000000000.0)
        (fma (- x_m) z x_m)
        (* (* z x_m) (- y 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 t_0 = (1.0 - y) * z;
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = z * ((y - 1.0) * x_m);
	} else if (t_0 <= 20000000000.0) {
		tmp = fma(-x_m, z, x_m);
	} else {
		tmp = (z * x_m) * (y - 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)
	t_0 = Float64(Float64(1.0 - y) * z)
	tmp = 0.0
	if (t_0 <= -200000.0)
		tmp = Float64(z * Float64(Float64(y - 1.0) * x_m));
	elseif (t_0 <= 20000000000.0)
		tmp = fma(Float64(-x_m), z, x_m);
	else
		tmp = Float64(Float64(z * x_m) * Float64(y - 1.0));
	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[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -200000.0], N[(z * N[(N[(y - 1.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 20000000000.0], N[((-x$95$m) * z + x$95$m), $MachinePrecision], N[(N[(z * x$95$m), $MachinePrecision] * N[(y - 1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) y) z) < -2e5

   1. Initial program 94.1%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. neg-mul-1 N/A

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

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

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 97.7%

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

   if -2e5 < (*.f64 (-.f64 #s(literal 1 binary64) y) z) < 2e10

   1. Initial program 100.0%

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

   4. Applied rewrites 93.1%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.2

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

   7. Applied rewrites 98.2%

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

   if 2e10 < (*.f64 (-.f64 #s(literal 1 binary64) y) z)

   1. Initial program 93.7%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. neg-mul-1 N/A

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

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

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 96.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \leq -200000:\\ \;\;\;\;z \cdot \left(\left(y - 1\right) \cdot x\right)\\ \mathbf{elif}\;\left(1 - y\right) \cdot z \leq 20000000000:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(y - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ t_1 := z \cdot \left(\left(y - 1\right) \cdot x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 20000000000:\\ \;\;\;\;\mathsf{fma}\left(-x\_m, z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* (- 1.0 y) z)) (t_1 (* z (* (- y 1.0) x_m))))
   (*
    x_s
    (if (<= t_0 -200000.0)
      t_1
      (if (<= t_0 20000000000.0) (fma (- x_m) z x_m) 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 = (1.0 - y) * z;
	double t_1 = z * ((y - 1.0) * x_m);
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = t_1;
	} else if (t_0 <= 20000000000.0) {
		tmp = fma(-x_m, z, x_m);
	} 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(Float64(1.0 - y) * z)
	t_1 = Float64(z * Float64(Float64(y - 1.0) * x_m))
	tmp = 0.0
	if (t_0 <= -200000.0)
		tmp = t_1;
	elseif (t_0 <= 20000000000.0)
		tmp = fma(Float64(-x_m), z, x_m);
	else
		tmp = t_1;
	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[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(N[(y - 1.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -200000.0], t$95$1, If[LessEqual[t$95$0, 20000000000.0], N[((-x$95$m) * z + x$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) y) z) < -2e5 or 2e10 < (*.f64 (-.f64 #s(literal 1 binary64) y) z)

   1. Initial program 93.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. neg-mul-1 N/A

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

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

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 95.9%

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

   if -2e5 < (*.f64 (-.f64 #s(literal 1 binary64) y) z) < 2e10

   1. Initial program 100.0%

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

   4. Applied rewrites 93.1%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.2

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

   7. Applied rewrites 98.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \leq -200000:\\ \;\;\;\;z \cdot \left(\left(y - 1\right) \cdot x\right)\\ \mathbf{elif}\;\left(1 - y\right) \cdot z \leq 20000000000:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y - 1\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(1 - \left(1 - y\right) \cdot z\right) \cdot x\_m \leq -5 \cdot 10^{+228}:\\ \;\;\;\;\left(z \cdot x\_m\right) \cdot \left(y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot x\_m, z, x\_m\right)\\ \end{array} \end{array} \]

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 (* (- 1.0 y) z)) x_m) -5e+228)
    (* (* z x_m) (- y 1.0))
    (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 (((1.0 - ((1.0 - y) * z)) * x_m) <= -5e+228) {
		tmp = (z * x_m) * (y - 1.0);
	} 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 (Float64(Float64(1.0 - Float64(Float64(1.0 - y) * z)) * x_m) <= -5e+228)
		tmp = Float64(Float64(z * x_m) * Float64(y - 1.0));
	else
		tmp = fma(Float64(Float64(y - 1.0) * 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[(N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], -5e+228], N[(N[(z * x$95$m), $MachinePrecision] * N[(y - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * z + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z))) < -5e228

   1. Initial program 85.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. neg-mul-1 N/A

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

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

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 89.8%

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

   if -5e228 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)))

   1. Initial program 98.5%

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

   4. Applied rewrites 95.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left(1 - y\right) \cdot z\right) \cdot x \leq -5 \cdot 10^{+228}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot x\_m, z, x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \leq -4 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 1.000005:\\ \;\;\;\;\mathsf{fma}\left(-x\_m, z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* y x_m) z x_m)))
   (*
    x_s
    (if (<= (- 1.0 y) -4e+26)
      t_0
      (if (<= (- 1.0 y) 1.000005) (fma (- x_m) z x_m) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = fma((y * x_m), z, x_m);
	double tmp;
	if ((1.0 - y) <= -4e+26) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1.000005) {
		tmp = fma(-x_m, z, x_m);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = fma(Float64(y * x_m), z, x_m)
	tmp = 0.0
	if (Float64(1.0 - y) <= -4e+26)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 1.000005)
		tmp = fma(Float64(-x_m), z, x_m);
	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), $MachinePrecision] * z + x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(1.0 - y), $MachinePrecision], -4e+26], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 1.000005], N[((-x$95$m) * z + x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -4.00000000000000019e26 or 1.00000500000000003 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 92.6%

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 90.5

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

   7. Applied rewrites 90.5%

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

   if -4.00000000000000019e26 < (-.f64 #s(literal 1 binary64) y) < 1.00000500000000003

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.5

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

   7. Applied rewrites 98.5%

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

Alternative 6: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(y \cdot x\_m\right) \cdot z\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;1 - y \leq -2 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 2 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(-x\_m, z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (* y x_m) z)))
   (*
    x_s
    (if (<= (- 1.0 y) -2e+51)
      t_0
      (if (<= (- 1.0 y) 2e+27) (fma (- x_m) z x_m) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (y * x_m) * z;
	double tmp;
	if ((1.0 - y) <= -2e+51) {
		tmp = t_0;
	} else if ((1.0 - y) <= 2e+27) {
		tmp = fma(-x_m, z, x_m);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(y * x_m) * z)
	tmp = 0.0
	if (Float64(1.0 - y) <= -2e+51)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 2e+27)
		tmp = fma(Float64(-x_m), z, x_m);
	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), $MachinePrecision] * z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(1.0 - y), $MachinePrecision], -2e+51], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 2e+27], N[((-x$95$m) * z + x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -2e51 or 2e27 < (-.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 94.0%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 72.5

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

   7. Applied rewrites 72.5%

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

   if -2e51 < (-.f64 #s(literal 1 binary64) y) < 2e27

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 94.7

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

   7. Applied rewrites 94.7%

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

Alternative 7: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(z \cdot x\_m\right) \cdot y\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(-x\_m, z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (* z x_m) y)))
   (*
    x_s
    (if (<= y -2.2e+32) t_0 (if (<= y 2.7e+47) (fma (- x_m) z x_m) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (z * x_m) * y;
	double tmp;
	if (y <= -2.2e+32) {
		tmp = t_0;
	} else if (y <= 2.7e+47) {
		tmp = fma(-x_m, z, x_m);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(z * x_m) * y)
	tmp = 0.0
	if (y <= -2.2e+32)
		tmp = t_0;
	elseif (y <= 2.7e+47)
		tmp = fma(Float64(-x_m), z, x_m);
	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[(z * x$95$m), $MachinePrecision] * y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -2.2e+32], t$95$0, If[LessEqual[y, 2.7e+47], N[((-x$95$m) * z + x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000001e32 or 2.69999999999999996e47 < y

   1. Initial program 91.5%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. lower-*.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if -2.20000000000000001e32 < y < 2.69999999999999996e47

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 94.7

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

   7. Applied rewrites 94.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+32}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(-x\_m\right) \cdot z\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 135000000:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (- x_m) z)))
   (* x_s (if (<= z -1.0) t_0 (if (<= z 135000000.0) (* 1.0 x_m) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = -x_m * z;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 135000000.0) {
		tmp = 1.0 * x_m;
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x_m * z
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 135000000.0d0) then
        tmp = 1.0d0 * x_m
    else
        tmp = t_0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = -x_m * z;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 135000000.0) {
		tmp = 1.0 * x_m;
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = -x_m * z
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 135000000.0:
		tmp = 1.0 * x_m
	else:
		tmp = t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(-x_m) * z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 135000000.0)
		tmp = Float64(1.0 * x_m);
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = -x_m * z;
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 135000000.0)
		tmp = 1.0 * x_m;
	else
		tmp = t_0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[((-x$95$m) * z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 135000000.0], N[(1.0 * x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1.35e8 < z

   1. Initial program 92.5%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. neg-mul-1 N/A

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

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

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 56.2%

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

   if -1 < z < 1.35e8

   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 73.9%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\left(-x\right) \cdot z\\ \mathbf{elif}\;z \leq 135000000:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.9% accurate, 1.9× speedup

Math

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

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

Derivation

1. Initial program 96.2%

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

4. Applied rewrites 95.2%

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

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 65.7

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

7. Applied rewrites 65.7%

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

Alternative 10: 65.9% accurate, 1.9× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (* (- 1.0 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 * ((1.0 - 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 * ((1.0d0 - 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 * ((1.0 - 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 * ((1.0 - 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(Float64(1.0 - 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 * ((1.0 - 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[(1.0 - z), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
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 65.7

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

5. Applied rewrites 65.7%

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

6. Final simplification 65.7%

   \[\leadsto \left(1 - z\right) \cdot x \]
7. Add Preprocessing

Alternative 11: 39.3% accurate, 2.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (* 1.0 x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (1.0 * x_m);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (1.0d0 * x_m)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (1.0 * x_m);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (1.0 * x_m)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(1.0 * x_m))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (1.0 * x_m);
end

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 38.8%

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

6. Final simplification 38.8%

   \[\leadsto 1 \cdot x \]
7. Add Preprocessing

Developer Target 1: 99.7% 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 2024244 
(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))))