Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D

Percentage Accurate: 99.5% → 99.5%

Time: 10.6s

Alternatives: 14

Speedup: 1.7×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (((y - x) * 6.0d0) * ((2.0d0 / 3.0d0) - z))
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * Float64(Float64(2.0 / 3.0) - z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (((y - x) * 6.0d0) * ((2.0d0 / 3.0d0) - z))
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * Float64(Float64(2.0 / 3.0) - z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (- 0.6666666666666666 z) (* 6.0 (- y x)) x))

C

double code(double x, double y, double z) {
	return fma((0.6666666666666666 - z), (6.0 * (y - x)), x);
}

Julia

function code(x, y, z)
	return fma(Float64(0.6666666666666666 - z), Float64(6.0 * Float64(y - x)), x)
end

Wolfram

code[x_, y_, z_] := N[(N[(0.6666666666666666 - z), $MachinePrecision] * N[(6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

   5. lower-fma.f64 99.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

   6. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

   7. metadata-eval 99.5

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   10. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]
5. Add Preprocessing

Alternative 2: 75.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (fma -6.0 z 4.0) y)))
   (if (<= t_0 -100.0)
     t_1
     (if (<= t_0 1.0)
       (fma -3.0 x (* 4.0 y))
       (if (<= t_0 5e+144) t_1 (* (* 6.0 x) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = fma(-6.0, z, 4.0) * y;
	double tmp;
	if (t_0 <= -100.0) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else if (t_0 <= 5e+144) {
		tmp = t_1;
	} else {
		tmp = (6.0 * x) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(fma(-6.0, z, 4.0) * y)
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	elseif (t_0 <= 5e+144)
		tmp = t_1;
	else
		tmp = Float64(Float64(6.0 * x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -100.0], t$95$1, If[LessEqual[t$95$0, 1.0], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+144], t$95$1, N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -100 or 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 4.9999999999999999e144

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      4. *-lft-identity N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right)\right) \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right)\right) \cdot y \]

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

         \[\leadsto \left(6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}\right) \cdot y \]

      7. +-commutative N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}\right) \cdot y \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot z\right) \cdot 6 + \frac{2}{3} \cdot 6\right)} \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot z\right) \cdot 6 + \color{blue}{4}\right) \cdot y \]

      10. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot 6 + 4\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z \cdot 6\right)\right)} + 4\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(6\right)\right)} + 4\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(z \cdot \color{blue}{-6} + 4\right) \cdot y \]

      14. *-commutative N/A

          \[\leadsto \left(\color{blue}{-6 \cdot z} + 4\right) \cdot y \]

      15. lower-fma.f64 57.5

          \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right)} \cdot y \]

   5. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right) \cdot y} \]

   if -100 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 99.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 99.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 99.3

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in z around 0

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

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

         \[\leadsto x + \color{blue}{\left(4 \cdot y - 4 \cdot x\right)} \]

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

         \[\leadsto x + \color{blue}{\left(4 \cdot y + \left(\mathsf{neg}\left(4\right)\right) \cdot x\right)} \]

      3. metadata-eval N/A

         \[\leadsto x + \left(4 \cdot y + \color{blue}{-4} \cdot x\right) \]

      4. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(-4 \cdot x + 4 \cdot y\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -4 \cdot x\right) + 4 \cdot y} \]

      6. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + 4 \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

      9. lower-*.f64 98.7

         \[\leadsto \mathsf{fma}\left(-3, x, \color{blue}{4 \cdot y}\right) \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

   if 4.9999999999999999e144 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 100.0%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(-6 \cdot \left(y - x\right) + \left(4 \cdot \frac{y - x}{z} + \frac{x}{z}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right) + \left(4 \cdot \frac{y - x}{z} + \frac{x}{z}\right)\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right) + \left(4 \cdot \frac{y - x}{z} + \frac{x}{z}\right)\right) \cdot z} \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(-6 \cdot \left(y - x\right) + 4 \cdot \frac{y - x}{z}\right) + \frac{x}{z}\right)} \cdot z \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(4 \cdot \frac{y - x}{z} + -6 \cdot \left(y - x\right)\right)} + \frac{x}{z}\right) \cdot z \]

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

         \[\leadsto \left(\color{blue}{\left(4 \cdot \frac{y - x}{z} - \left(\mathsf{neg}\left(-6\right)\right) \cdot \left(y - x\right)\right)} + \frac{x}{z}\right) \cdot z \]

      6. associate-*r/ N/A

         \[\leadsto \left(\left(\color{blue}{\frac{4 \cdot \left(y - x\right)}{z}} - \left(\mathsf{neg}\left(-6\right)\right) \cdot \left(y - x\right)\right) + \frac{x}{z}\right) \cdot z \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(y - x\right) \cdot 4}}{z} - \left(\mathsf{neg}\left(-6\right)\right) \cdot \left(y - x\right)\right) + \frac{x}{z}\right) \cdot z \]

      8. associate-/l* N/A

         \[\leadsto \left(\left(\color{blue}{\left(y - x\right) \cdot \frac{4}{z}} - \left(\mathsf{neg}\left(-6\right)\right) \cdot \left(y - x\right)\right) + \frac{x}{z}\right) \cdot z \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(y - x\right) \cdot \frac{4}{z} - \color{blue}{6} \cdot \left(y - x\right)\right) + \frac{x}{z}\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\left(y - x\right) \cdot \frac{4}{z} - \color{blue}{\left(y - x\right) \cdot 6}\right) + \frac{x}{z}\right) \cdot z \]

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

          \[\leadsto \left(\color{blue}{\left(y - x\right) \cdot \left(\frac{4}{z} - 6\right)} + \frac{x}{z}\right) \cdot z \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{4}{z} - 6, \frac{x}{z}\right)} \cdot z \]

      13. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{4}{z} - 6, \frac{x}{z}\right) \cdot z \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{4}{z} - 6}, \frac{x}{z}\right) \cdot z \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{4}{z}} - 6, \frac{x}{z}\right) \cdot z \]

      16. lower-/.f64 99.9

          \[\leadsto \mathsf{fma}\left(y - x, \frac{4}{z} - 6, \color{blue}{\frac{x}{z}}\right) \cdot z \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{4}{z} - 6, \frac{x}{z}\right) \cdot z} \]

   8. Taylor expanded in z around inf

      \[\leadsto \left(-6 \cdot \left(y - x\right)\right) \cdot z \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \left(-6 \cdot \left(y - x\right)\right) \cdot z \]

   11. Taylor expanded in x around inf

       \[\leadsto \left(6 \cdot x\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 67.1%

       \[\leadsto \left(6 \cdot x\right) \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -100:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{elif}\;\frac{2}{3} - z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (fma -6.0 z 4.0) y)))
   (if (<= t_0 -100.0)
     t_1
     (if (<= t_0 1.0)
       (fma (- y x) 4.0 x)
       (if (<= t_0 5e+144) t_1 (* (* 6.0 x) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = fma(-6.0, z, 4.0) * y;
	double tmp;
	if (t_0 <= -100.0) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = fma((y - x), 4.0, x);
	} else if (t_0 <= 5e+144) {
		tmp = t_1;
	} else {
		tmp = (6.0 * x) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(fma(-6.0, z, 4.0) * y)
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = fma(Float64(y - x), 4.0, x);
	elseif (t_0 <= 5e+144)
		tmp = t_1;
	else
		tmp = Float64(Float64(6.0 * x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -100.0], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], If[LessEqual[t$95$0, 5e+144], t$95$1, N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -100 or 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 4.9999999999999999e144

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      4. *-lft-identity N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right)\right) \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right)\right) \cdot y \]

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

         \[\leadsto \left(6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}\right) \cdot y \]

      7. +-commutative N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}\right) \cdot y \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot z\right) \cdot 6 + \frac{2}{3} \cdot 6\right)} \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot z\right) \cdot 6 + \color{blue}{4}\right) \cdot y \]

      10. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot 6 + 4\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z \cdot 6\right)\right)} + 4\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(6\right)\right)} + 4\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(z \cdot \color{blue}{-6} + 4\right) \cdot y \]

      14. *-commutative N/A

          \[\leadsto \left(\color{blue}{-6 \cdot z} + 4\right) \cdot y \]

      15. lower-fma.f64 57.5

          \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right)} \cdot y \]

   5. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right) \cdot y} \]

   if -100 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 98.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   if 4.9999999999999999e144 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 100.0%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(-6 \cdot \left(y - x\right) + \left(4 \cdot \frac{y - x}{z} + \frac{x}{z}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right) + \left(4 \cdot \frac{y - x}{z} + \frac{x}{z}\right)\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right) + \left(4 \cdot \frac{y - x}{z} + \frac{x}{z}\right)\right) \cdot z} \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(-6 \cdot \left(y - x\right) + 4 \cdot \frac{y - x}{z}\right) + \frac{x}{z}\right)} \cdot z \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(4 \cdot \frac{y - x}{z} + -6 \cdot \left(y - x\right)\right)} + \frac{x}{z}\right) \cdot z \]

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

         \[\leadsto \left(\color{blue}{\left(4 \cdot \frac{y - x}{z} - \left(\mathsf{neg}\left(-6\right)\right) \cdot \left(y - x\right)\right)} + \frac{x}{z}\right) \cdot z \]

      6. associate-*r/ N/A

         \[\leadsto \left(\left(\color{blue}{\frac{4 \cdot \left(y - x\right)}{z}} - \left(\mathsf{neg}\left(-6\right)\right) \cdot \left(y - x\right)\right) + \frac{x}{z}\right) \cdot z \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(y - x\right) \cdot 4}}{z} - \left(\mathsf{neg}\left(-6\right)\right) \cdot \left(y - x\right)\right) + \frac{x}{z}\right) \cdot z \]

      8. associate-/l* N/A

         \[\leadsto \left(\left(\color{blue}{\left(y - x\right) \cdot \frac{4}{z}} - \left(\mathsf{neg}\left(-6\right)\right) \cdot \left(y - x\right)\right) + \frac{x}{z}\right) \cdot z \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(y - x\right) \cdot \frac{4}{z} - \color{blue}{6} \cdot \left(y - x\right)\right) + \frac{x}{z}\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\left(y - x\right) \cdot \frac{4}{z} - \color{blue}{\left(y - x\right) \cdot 6}\right) + \frac{x}{z}\right) \cdot z \]

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

          \[\leadsto \left(\color{blue}{\left(y - x\right) \cdot \left(\frac{4}{z} - 6\right)} + \frac{x}{z}\right) \cdot z \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{4}{z} - 6, \frac{x}{z}\right)} \cdot z \]

      13. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{4}{z} - 6, \frac{x}{z}\right) \cdot z \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{4}{z} - 6}, \frac{x}{z}\right) \cdot z \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{4}{z}} - 6, \frac{x}{z}\right) \cdot z \]

      16. lower-/.f64 99.9

          \[\leadsto \mathsf{fma}\left(y - x, \frac{4}{z} - 6, \color{blue}{\frac{x}{z}}\right) \cdot z \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{4}{z} - 6, \frac{x}{z}\right) \cdot z} \]

   8. Taylor expanded in z around inf

      \[\leadsto \left(-6 \cdot \left(y - x\right)\right) \cdot z \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \left(-6 \cdot \left(y - x\right)\right) \cdot z \]

   11. Taylor expanded in x around inf

       \[\leadsto \left(6 \cdot x\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 67.1%

       \[\leadsto \left(6 \cdot x\right) \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -100:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{elif}\;\frac{2}{3} - z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -100 \lor \neg \left(t\_0 \leq 1000\right):\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (or (<= t_0 -100.0) (not (<= t_0 1000.0)))
     (* (* x z) 6.0)
     (fma (- y x) 4.0 x))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if ((t_0 <= -100.0) || !(t_0 <= 1000.0)) {
		tmp = (x * z) * 6.0;
	} else {
		tmp = fma((y - x), 4.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if ((t_0 <= -100.0) || !(t_0 <= 1000.0))
		tmp = Float64(Float64(x * z) * 6.0);
	else
		tmp = fma(Float64(y - x), 4.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -100.0], N[Not[LessEqual[t$95$0, 1000.0]], $MachinePrecision]], N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -100 or 1e3 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      5. lower--.f64 98.4

         \[\leadsto \left(\color{blue}{\left(y - x\right)} \cdot z\right) \cdot -6 \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 52.5%

      \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{6} \]

   if -100 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1e3

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 97.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 97.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -100 \lor \neg \left(\frac{2}{3} - z \leq 1000\right):\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.56 \lor \neg \left(z \leq 0.55\right):\\ \;\;\;\;\left(-6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.56) (not (<= z 0.55)))
   (* (* -6.0 (- y x)) z)
   (fma -3.0 x (* 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.56) || !(z <= 0.55)) {
		tmp = (-6.0 * (y - x)) * z;
	} else {
		tmp = fma(-3.0, x, (4.0 * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.56) || !(z <= 0.55))
		tmp = Float64(Float64(-6.0 * Float64(y - x)) * z);
	else
		tmp = fma(-3.0, x, Float64(4.0 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.56], N[Not[LessEqual[z, 0.55]], $MachinePrecision]], N[(N[(-6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.56000000000000005 or 0.55000000000000004 < z

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      5. lower--.f64 97.9

         \[\leadsto \left(\color{blue}{\left(y - x\right)} \cdot z\right) \cdot -6 \]

   5. Applied rewrites 97.9%

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

   7. Applied rewrites 97.9%

      \[\leadsto \left(-6 \cdot \left(y - x\right)\right) \cdot \color{blue}{z} \]

   if -0.56000000000000005 < z < 0.55000000000000004

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 99.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 99.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 99.3

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in z around 0

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

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

         \[\leadsto x + \color{blue}{\left(4 \cdot y - 4 \cdot x\right)} \]

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

         \[\leadsto x + \color{blue}{\left(4 \cdot y + \left(\mathsf{neg}\left(4\right)\right) \cdot x\right)} \]

      3. metadata-eval N/A

         \[\leadsto x + \left(4 \cdot y + \color{blue}{-4} \cdot x\right) \]

      4. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(-4 \cdot x + 4 \cdot y\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -4 \cdot x\right) + 4 \cdot y} \]

      6. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + 4 \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

      9. lower-*.f64 98.1

         \[\leadsto \mathsf{fma}\left(-3, x, \color{blue}{4 \cdot y}\right) \]

   7. Applied rewrites 98.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.56 \lor \neg \left(z \leq 0.55\right):\\ \;\;\;\;\left(-6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.56:\\ \;\;\;\;\left(-6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{elif}\;z \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.56)
   (* (* -6.0 (- y x)) z)
   (if (<= z 0.55) (fma -3.0 x (* 4.0 y)) (* (* (- y x) z) -6.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.56) {
		tmp = (-6.0 * (y - x)) * z;
	} else if (z <= 0.55) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else {
		tmp = ((y - x) * z) * -6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.56)
		tmp = Float64(Float64(-6.0 * Float64(y - x)) * z);
	elseif (z <= 0.55)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	else
		tmp = Float64(Float64(Float64(y - x) * z) * -6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.56], N[(N[(-6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 0.55], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * -6.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.56000000000000005

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      5. lower--.f64 98.4

         \[\leadsto \left(\color{blue}{\left(y - x\right)} \cdot z\right) \cdot -6 \]

   5. Applied rewrites 98.4%

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

   7. Applied rewrites 98.5%

      \[\leadsto \left(-6 \cdot \left(y - x\right)\right) \cdot \color{blue}{z} \]

   if -0.56000000000000005 < z < 0.55000000000000004

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 99.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 99.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 99.3

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in z around 0

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

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

         \[\leadsto x + \color{blue}{\left(4 \cdot y - 4 \cdot x\right)} \]

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

         \[\leadsto x + \color{blue}{\left(4 \cdot y + \left(\mathsf{neg}\left(4\right)\right) \cdot x\right)} \]

      3. metadata-eval N/A

         \[\leadsto x + \left(4 \cdot y + \color{blue}{-4} \cdot x\right) \]

      4. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(-4 \cdot x + 4 \cdot y\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -4 \cdot x\right) + 4 \cdot y} \]

      6. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + 4 \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

      9. lower-*.f64 98.1

         \[\leadsto \mathsf{fma}\left(-3, x, \color{blue}{4 \cdot y}\right) \]

   7. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

   if 0.55000000000000004 < z

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      5. lower--.f64 97.2

         \[\leadsto \left(\color{blue}{\left(y - x\right)} \cdot z\right) \cdot -6 \]

   5. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.56:\\ \;\;\;\;\left(-6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{elif}\;z \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-66} \lor \neg \left(x \leq 1.65 \cdot 10^{-32}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.7e-66) (not (<= x 1.65e-32)))
   (fma (fma 6.0 z -4.0) x x)
   (* (fma -6.0 z 4.0) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.7e-66) || !(x <= 1.65e-32)) {
		tmp = fma(fma(6.0, z, -4.0), x, x);
	} else {
		tmp = fma(-6.0, z, 4.0) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.7e-66) || !(x <= 1.65e-32))
		tmp = fma(fma(6.0, z, -4.0), x, x);
	else
		tmp = Float64(fma(-6.0, z, 4.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.7e-66], N[Not[LessEqual[x, 1.65e-32]], $MachinePrecision]], N[(N[(6.0 * z + -4.0), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7000000000000002e-66 or 1.65000000000000013e-32 < x

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 76.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]

   if -3.7000000000000002e-66 < x < 1.65000000000000013e-32

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      4. *-lft-identity N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right)\right) \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right)\right) \cdot y \]

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

         \[\leadsto \left(6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}\right) \cdot y \]

      7. +-commutative N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}\right) \cdot y \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot z\right) \cdot 6 + \frac{2}{3} \cdot 6\right)} \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot z\right) \cdot 6 + \color{blue}{4}\right) \cdot y \]

      10. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot 6 + 4\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z \cdot 6\right)\right)} + 4\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(6\right)\right)} + 4\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(z \cdot \color{blue}{-6} + 4\right) \cdot y \]

      14. *-commutative N/A

          \[\leadsto \left(\color{blue}{-6 \cdot z} + 4\right) \cdot y \]

      15. lower-fma.f64 83.8

          \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right)} \cdot y \]

   5. Applied rewrites 83.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-66} \lor \neg \left(x \leq 1.65 \cdot 10^{-32}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -660:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -660.0)
   (* (* 6.0 x) z)
   (if (<= z 0.68) (fma (- y x) 4.0 x) (* (* -6.0 z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -660.0) {
		tmp = (6.0 * x) * z;
	} else if (z <= 0.68) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = (-6.0 * z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -660.0)
		tmp = Float64(Float64(6.0 * x) * z);
	elseif (z <= 0.68)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = Float64(Float64(-6.0 * z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -660.0], N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 0.68], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], N[(N[(-6.0 * z), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -660

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 99.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 99.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(-6 \cdot \left(y - x\right) + \left(4 \cdot \frac{y - x}{z} + \frac{x}{z}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right) + \left(4 \cdot \frac{y - x}{z} + \frac{x}{z}\right)\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(y - x\right) + \left(4 \cdot \frac{y - x}{z} + \frac{x}{z}\right)\right) \cdot z} \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(-6 \cdot \left(y - x\right) + 4 \cdot \frac{y - x}{z}\right) + \frac{x}{z}\right)} \cdot z \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(4 \cdot \frac{y - x}{z} + -6 \cdot \left(y - x\right)\right)} + \frac{x}{z}\right) \cdot z \]

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

         \[\leadsto \left(\color{blue}{\left(4 \cdot \frac{y - x}{z} - \left(\mathsf{neg}\left(-6\right)\right) \cdot \left(y - x\right)\right)} + \frac{x}{z}\right) \cdot z \]

      6. associate-*r/ N/A

         \[\leadsto \left(\left(\color{blue}{\frac{4 \cdot \left(y - x\right)}{z}} - \left(\mathsf{neg}\left(-6\right)\right) \cdot \left(y - x\right)\right) + \frac{x}{z}\right) \cdot z \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(y - x\right) \cdot 4}}{z} - \left(\mathsf{neg}\left(-6\right)\right) \cdot \left(y - x\right)\right) + \frac{x}{z}\right) \cdot z \]

      8. associate-/l* N/A

         \[\leadsto \left(\left(\color{blue}{\left(y - x\right) \cdot \frac{4}{z}} - \left(\mathsf{neg}\left(-6\right)\right) \cdot \left(y - x\right)\right) + \frac{x}{z}\right) \cdot z \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(y - x\right) \cdot \frac{4}{z} - \color{blue}{6} \cdot \left(y - x\right)\right) + \frac{x}{z}\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\left(y - x\right) \cdot \frac{4}{z} - \color{blue}{\left(y - x\right) \cdot 6}\right) + \frac{x}{z}\right) \cdot z \]

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

          \[\leadsto \left(\color{blue}{\left(y - x\right) \cdot \left(\frac{4}{z} - 6\right)} + \frac{x}{z}\right) \cdot z \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{4}{z} - 6, \frac{x}{z}\right)} \cdot z \]

      13. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{4}{z} - 6, \frac{x}{z}\right) \cdot z \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{4}{z} - 6}, \frac{x}{z}\right) \cdot z \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{4}{z}} - 6, \frac{x}{z}\right) \cdot z \]

      16. lower-/.f64 99.9

          \[\leadsto \mathsf{fma}\left(y - x, \frac{4}{z} - 6, \color{blue}{\frac{x}{z}}\right) \cdot z \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{4}{z} - 6, \frac{x}{z}\right) \cdot z} \]

   8. Taylor expanded in z around inf

      \[\leadsto \left(-6 \cdot \left(y - x\right)\right) \cdot z \]
   9. Step-by-step derivation

   10. Applied rewrites 99.4%

       \[\leadsto \left(-6 \cdot \left(y - x\right)\right) \cdot z \]

   11. Taylor expanded in x around inf

       \[\leadsto \left(6 \cdot x\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 56.6%

       \[\leadsto \left(6 \cdot x\right) \cdot z \]

   if -660 < z < 0.680000000000000049

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 97.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   if 0.680000000000000049 < z

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 99.6

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 99.6

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot \left(\frac{2}{3} - z\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      4. *-lft-identity N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right)\right) \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right)\right) \cdot y \]

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

         \[\leadsto \left(6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}\right) \cdot y \]

      7. +-commutative N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}\right) \cdot y \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(-1 \cdot z\right) + 6 \cdot \frac{2}{3}\right)} \cdot y \]

      9. mul-1-neg N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + 6 \cdot \frac{2}{3}\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(6 \cdot z\right)\right)} + 6 \cdot \frac{2}{3}\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(6\right)\right) \cdot z} + 6 \cdot \frac{2}{3}\right) \cdot y \]

      12. metadata-eval N/A

          \[\leadsto \left(\color{blue}{-6} \cdot z + 6 \cdot \frac{2}{3}\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(-6 \cdot z + \color{blue}{4}\right) \cdot y \]

      14. lower-fma.f64 57.3

          \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right)} \cdot y \]

   7. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right) \cdot y} \]

   8. Taylor expanded in z around 0

      \[\leadsto 4 \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 1.4%

       \[\leadsto 4 \cdot y \]

   11. Taylor expanded in z around inf

       \[\leadsto \left(-6 \cdot z\right) \cdot y \]
   12. Step-by-step derivation

   13. Applied rewrites 56.0%

       \[\leadsto \left(-6 \cdot z\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -660:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -660:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -660.0)
   (* (* x z) 6.0)
   (if (<= z 0.68) (fma (- y x) 4.0 x) (* (* -6.0 z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -660.0) {
		tmp = (x * z) * 6.0;
	} else if (z <= 0.68) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = (-6.0 * z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -660.0)
		tmp = Float64(Float64(x * z) * 6.0);
	elseif (z <= 0.68)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = Float64(Float64(-6.0 * z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -660.0], N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision], If[LessEqual[z, 0.68], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], N[(N[(-6.0 * z), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -660

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      5. lower--.f64 99.3

         \[\leadsto \left(\color{blue}{\left(y - x\right)} \cdot z\right) \cdot -6 \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.5%

      \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{6} \]

   if -660 < z < 0.680000000000000049

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 97.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   if 0.680000000000000049 < z

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 99.6

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 99.6

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot \left(\frac{2}{3} - z\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      4. *-lft-identity N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right)\right) \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right)\right) \cdot y \]

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

         \[\leadsto \left(6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}\right) \cdot y \]

      7. +-commutative N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}\right) \cdot y \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(-1 \cdot z\right) + 6 \cdot \frac{2}{3}\right)} \cdot y \]

      9. mul-1-neg N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + 6 \cdot \frac{2}{3}\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(6 \cdot z\right)\right)} + 6 \cdot \frac{2}{3}\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(6\right)\right) \cdot z} + 6 \cdot \frac{2}{3}\right) \cdot y \]

      12. metadata-eval N/A

          \[\leadsto \left(\color{blue}{-6} \cdot z + 6 \cdot \frac{2}{3}\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(-6 \cdot z + \color{blue}{4}\right) \cdot y \]

      14. lower-fma.f64 57.3

          \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right)} \cdot y \]

   7. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right) \cdot y} \]

   8. Taylor expanded in z around 0

      \[\leadsto 4 \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 1.4%

       \[\leadsto 4 \cdot y \]

   11. Taylor expanded in z around inf

       \[\leadsto \left(-6 \cdot z\right) \cdot y \]
   12. Step-by-step derivation

   13. Applied rewrites 56.0%

       \[\leadsto \left(-6 \cdot z\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -660:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -660:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -660.0)
   (* (* x z) 6.0)
   (if (<= z 0.68) (fma (- y x) 4.0 x) (* (* y z) -6.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -660.0) {
		tmp = (x * z) * 6.0;
	} else if (z <= 0.68) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = (y * z) * -6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -660.0)
		tmp = Float64(Float64(x * z) * 6.0);
	elseif (z <= 0.68)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = Float64(Float64(y * z) * -6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -660.0], N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision], If[LessEqual[z, 0.68], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * -6.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -660

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      5. lower--.f64 99.3

         \[\leadsto \left(\color{blue}{\left(y - x\right)} \cdot z\right) \cdot -6 \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.5%

      \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{6} \]

   if -660 < z < 0.680000000000000049

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 97.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   if 0.680000000000000049 < z

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot -6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot -6 \]

      5. lower--.f64 97.2

         \[\leadsto \left(\color{blue}{\left(y - x\right)} \cdot z\right) \cdot -6 \]

   5. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(y \cdot z\right) \cdot -6 \]
   7. Step-by-step derivation

   8. Applied rewrites 55.9%

      \[\leadsto \left(y \cdot z\right) \cdot -6 \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -660:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-68} \lor \neg \left(x \leq 1.08 \cdot 10^{-32}\right):\\ \;\;\;\;-3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.5e-68) (not (<= x 1.08e-32))) (* -3.0 x) (* 4.0 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.5e-68) || !(x <= 1.08e-32)) {
		tmp = -3.0 * x;
	} else {
		tmp = 4.0 * y;
	}
	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) :: tmp
    if ((x <= (-5.5d-68)) .or. (.not. (x <= 1.08d-32))) then
        tmp = (-3.0d0) * x
    else
        tmp = 4.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.5e-68) || !(x <= 1.08e-32)) {
		tmp = -3.0 * x;
	} else {
		tmp = 4.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.5e-68) or not (x <= 1.08e-32):
		tmp = -3.0 * x
	else:
		tmp = 4.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.5e-68) || !(x <= 1.08e-32))
		tmp = Float64(-3.0 * x);
	else
		tmp = Float64(4.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.5e-68) || ~((x <= 1.08e-32)))
		tmp = -3.0 * x;
	else
		tmp = 4.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.5e-68], N[Not[LessEqual[x, 1.08e-32]], $MachinePrecision]], N[(-3.0 * x), $MachinePrecision], N[(4.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5000000000000003e-68 or 1.08e-32 < x

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 50.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto -3 \cdot \color{blue}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.7%

      \[\leadsto -3 \cdot \color{blue}{x} \]

   if -5.5000000000000003e-68 < x < 1.08e-32

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 55.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 55.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.3%

      \[\leadsto 4 \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-68} \lor \neg \left(x \leq 1.08 \cdot 10^{-32}\right):\\ \;\;\;\;-3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(0.6666666666666666 - z\right) \cdot \left(y - x\right), 6, x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (* (- 0.6666666666666666 z) (- y x)) 6.0 x))

C

double code(double x, double y, double z) {
	return fma(((0.6666666666666666 - z) * (y - x)), 6.0, x);
}

Julia

function code(x, y, z)
	return fma(Float64(Float64(0.6666666666666666 - z) * Float64(y - x)), 6.0, x)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(0.6666666666666666 - z), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] * 6.0 + x), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \left(\frac{2}{3} - z\right) + x \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x \]

   6. *-commutative N/A

      \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right)} + x \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot 6} + x \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot \left(\frac{2}{3} - z\right), 6, x\right)} \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(y - x\right)}, 6, x\right) \]

   10. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(y - x\right)}, 6, x\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{2}{3}} - z\right) \cdot \left(y - x\right), 6, x\right) \]

   12. metadata-eval 99.5

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{0.6666666666666666} - z\right) \cdot \left(y - x\right), 6, x\right) \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.6666666666666666 - z\right) \cdot \left(y - x\right), 6, x\right)} \]
5. Add Preprocessing

Alternative 13: 50.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - x, 4, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- y x) 4.0 x))

C

double code(double x, double y, double z) {
	return fma((y - x), 4.0, x);
}

Julia

function code(x, y, z)
	return fma(Float64(y - x), 4.0, x)
end

Wolfram

code[x_, y_, z_] := N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   4. lower--.f64 52.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

5. Applied rewrites 52.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

6. Final simplification 52.2%

   \[\leadsto \mathsf{fma}\left(y - x, 4, x\right) \]
7. Add Preprocessing

Alternative 14: 26.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ -3 \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* -3.0 x))

C

double code(double x, double y, double z) {
	return -3.0 * x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (-3.0d0) * x
end function

Java

public static double code(double x, double y, double z) {
	return -3.0 * x;
}

Python

def code(x, y, z):
	return -3.0 * x

Julia

function code(x, y, z)
	return Float64(-3.0 * x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -3.0 * x;
end

Wolfram

code[x_, y_, z_] := N[(-3.0 * x), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   4. lower--.f64 52.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

5. Applied rewrites 52.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto -3 \cdot \color{blue}{x} \]
7. Step-by-step derivation

8. Applied rewrites 27.6%

   \[\leadsto -3 \cdot \color{blue}{x} \]

9. Final simplification 27.6%

   \[\leadsto -3 \cdot x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  :precision binary64
  (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))