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

Percentage Accurate: 99.5% → 99.5%

Time: 15.0s

Alternatives: 11

Speedup: N/A×

• 20.6% 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, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. neg-mul-1 N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 2: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := y \cdot \left(-6 \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.6666:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* y (* -6.0 z))))
   (if (<= t_0 -1e+129)
     t_1
     (if (<= t_0 0.6666)
       (* x (fma 6.0 z -3.0))
       (if (<= t_0 1.0) (fma 4.0 (- y x) x) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = y * (-6.0 * z);
	double tmp;
	if (t_0 <= -1e+129) {
		tmp = t_1;
	} else if (t_0 <= 0.6666) {
		tmp = x * fma(6.0, z, -3.0);
	} else if (t_0 <= 1.0) {
		tmp = fma(4.0, (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(y * Float64(-6.0 * z))
	tmp = 0.0
	if (t_0 <= -1e+129)
		tmp = t_1;
	elseif (t_0 <= 0.6666)
		tmp = Float64(x * fma(6.0, z, -3.0));
	elseif (t_0 <= 1.0)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = t_1;
	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[(y * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+129], t$95$1, If[LessEqual[t$95$0, 0.6666], N[(x * N[(6.0 * z + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e129 or 1 < (-.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. 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. lower-*.f64 99.7

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

      10. lift-/.f64 N/A

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

      11. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

      16. lower-fma.f64 64.4

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 64.1%

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

   if -1e129 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.66659999999999997

   1. Initial program 99.4%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   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. remove-double-neg N/A

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

      2. neg-mul-1 N/A

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

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

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

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

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. neg-mul-1 N/A

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

      15. lower-*.f64 N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-in N/A

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

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

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

      20. metadata-eval N/A

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

   5. Applied rewrites 62.9%

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

   if 0.66659999999999997 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   1. Initial program 99.4%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   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. lower-fma.f64 N/A

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

      3. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 3: 97.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(z * Float64(6.0 * Float64(x - y)))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = fma(x, -3.0, Float64(y * 4.0));
	else
		tmp = t_1;
	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[(z * N[(6.0 * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], t$95$1, If[LessEqual[t$95$0, 1.0], N[(x * -3.0 + N[(y * 4.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

   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. 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. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.4444444444444444 - z \cdot z, \frac{1}{0.6666666666666666 + z} \cdot \left(\left(y - x\right) \cdot 6\right), x\right)} \]

   5. Taylor expanded in z around inf

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

      1. metadata-eval N/A

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

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

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. neg-sub0 N/A

         \[\leadsto 6 \cdot \left(z \cdot \color{blue}{\left(0 - \left(y - x\right)\right)}\right) \]

      9. associate-+l- N/A

         \[\leadsto 6 \cdot \left(z \cdot \color{blue}{\left(\left(0 - y\right) + x\right)}\right) \]

      10. neg-sub0 N/A

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

      11. neg-mul-1 N/A

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

      12. +-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. sub-neg N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 97.2

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

   7. Applied rewrites 97.2%

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

   9. Applied rewrites 97.3%

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

   if -2 < (-.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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. neg-mul-1 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 97.2%

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

   10. Taylor expanded in z around 0

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

       1. distribute-lft-in N/A

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

       2. associate-+r+ N/A

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

       3. *-lft-identity N/A

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

       4. associate-*r* N/A

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

       5. metadata-eval N/A

          \[\leadsto \left(1 \cdot x + \color{blue}{-4} \cdot x\right) + \frac{2}{3} \cdot \left(6 \cdot y\right) \]

       6. distribute-rgt-out N/A

          \[\leadsto \color{blue}{x \cdot \left(1 + -4\right)} + \frac{2}{3} \cdot \left(6 \cdot y\right) \]

       7. associate-*r* N/A

          \[\leadsto x \cdot \left(1 + -4\right) + \color{blue}{\left(\frac{2}{3} \cdot 6\right) \cdot y} \]

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. metadata-eval N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 97.7

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

   12. Applied rewrites 97.7%

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

4. Final simplification 97.5%

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

Alternative 4: 74.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(y * Float64(-6.0 * z))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = t_1;
	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[(y * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], t$95$1, If[LessEqual[t$95$0, 1.0], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

   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. 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. lower-*.f64 99.6

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

      10. lift-/.f64 N/A

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

      11. metadata-eval 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

      16. lower-fma.f64 60.3

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 59.0%

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

   if -2 < (-.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. lower-fma.f64 N/A

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

      3. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 5: 97.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(z \cdot \left(x - y\right)\right)\\ \mathbf{if}\;z \leq -0.58:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(x, -3, y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 6.0 (* z (- x y)))))
   (if (<= z -0.58) t_0 (if (<= z 0.5) (fma x -3.0 (* y 4.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 6.0 * (z * (x - y));
	double tmp;
	if (z <= -0.58) {
		tmp = t_0;
	} else if (z <= 0.5) {
		tmp = fma(x, -3.0, (y * 4.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(6.0 * Float64(z * Float64(x - y)))
	tmp = 0.0
	if (z <= -0.58)
		tmp = t_0;
	elseif (z <= 0.5)
		tmp = fma(x, -3.0, Float64(y * 4.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(6.0 * N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.58], t$95$0, If[LessEqual[z, 0.5], N[(x * -3.0 + N[(y * 4.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.57999999999999996 or 0.5 < z

   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 z around inf

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

      1. metadata-eval N/A

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

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

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

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

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

      4. lower-*.f64 N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. neg-mul-1 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. mul-1-neg N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if -0.57999999999999996 < z < 0.5

   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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. neg-mul-1 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 97.2%

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

   10. Taylor expanded in z around 0

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

       1. distribute-lft-in N/A

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

       2. associate-+r+ N/A

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

       3. *-lft-identity N/A

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

       4. associate-*r* N/A

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

       5. metadata-eval N/A

          \[\leadsto \left(1 \cdot x + \color{blue}{-4} \cdot x\right) + \frac{2}{3} \cdot \left(6 \cdot y\right) \]

       6. distribute-rgt-out N/A

          \[\leadsto \color{blue}{x \cdot \left(1 + -4\right)} + \frac{2}{3} \cdot \left(6 \cdot y\right) \]

       7. associate-*r* N/A

          \[\leadsto x \cdot \left(1 + -4\right) + \color{blue}{\left(\frac{2}{3} \cdot 6\right) \cdot y} \]

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. metadata-eval N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 97.7

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

   12. Applied rewrites 97.7%

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

Alternative 6: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \mathsf{fma}\left(z, -6, 4\right)\\ \mathbf{if}\;y \leq -115:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 190000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (fma z -6.0 4.0))))
   (if (<= y -115.0)
     t_0
     (if (<= y 190000000000.0) (* x (fma 6.0 z -3.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * fma(z, -6.0, 4.0);
	double tmp;
	if (y <= -115.0) {
		tmp = t_0;
	} else if (y <= 190000000000.0) {
		tmp = x * fma(6.0, z, -3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * fma(z, -6.0, 4.0))
	tmp = 0.0
	if (y <= -115.0)
		tmp = t_0;
	elseif (y <= 190000000000.0)
		tmp = Float64(x * fma(6.0, z, -3.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(z * -6.0 + 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -115.0], t$95$0, If[LessEqual[y, 190000000000.0], N[(x * N[(6.0 * z + -3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -115 or 1.9e11 < y

   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 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if -115 < y < 1.9e11

   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 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. remove-double-neg N/A

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

      2. neg-mul-1 N/A

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

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

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

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

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. neg-mul-1 N/A

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

      15. lower-*.f64 N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-in N/A

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

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

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

      20. metadata-eval N/A

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

   5. Applied rewrites 82.0%

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

Alternative 7: 38.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+18}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-11}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.2e+18) (* y 4.0) (if (<= y 4.4e-11) (* x -3.0) (* y 4.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.2e+18) {
		tmp = y * 4.0;
	} else if (y <= 4.4e-11) {
		tmp = x * -3.0;
	} else {
		tmp = y * 4.0;
	}
	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 (y <= (-4.2d+18)) then
        tmp = y * 4.0d0
    else if (y <= 4.4d-11) then
        tmp = x * (-3.0d0)
    else
        tmp = y * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.2e+18) {
		tmp = y * 4.0;
	} else if (y <= 4.4e-11) {
		tmp = x * -3.0;
	} else {
		tmp = y * 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.2e+18:
		tmp = y * 4.0
	elif y <= 4.4e-11:
		tmp = x * -3.0
	else:
		tmp = y * 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.2e+18)
		tmp = Float64(y * 4.0);
	elseif (y <= 4.4e-11)
		tmp = Float64(x * -3.0);
	else
		tmp = Float64(y * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.2e+18)
		tmp = y * 4.0;
	elseif (y <= 4.4e-11)
		tmp = x * -3.0;
	else
		tmp = y * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.2e+18], N[(y * 4.0), $MachinePrecision], If[LessEqual[y, 4.4e-11], N[(x * -3.0), $MachinePrecision], N[(y * 4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e18 or 4.4000000000000003e-11 < y

   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. lower-fma.f64 N/A

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

      3. lower--.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 42.3%

      \[\leadsto y \cdot \color{blue}{4} \]

   if -4.2e18 < y < 4.4000000000000003e-11

   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. lower-fma.f64 N/A

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

      3. lower--.f64 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 47.3%

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

Alternative 8: 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

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 9: 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. lower-*.f64 99.5

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

   10. lift-/.f64 N/A

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

   11. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 10: 51.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(4.0 * N[(y - x), $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. 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. lower-fma.f64 N/A

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

   3. lower--.f64 56.0

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

5. Applied rewrites 56.0%

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

Alternative 11: 26.6% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

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 * (-3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * -3.0), $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. lower-fma.f64 N/A

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

   3. lower--.f64 56.0

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

5. Applied rewrites 56.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 26.8%

   \[\leadsto x \cdot \color{blue}{-3} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(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))))