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

Percentage Accurate: 99.5% → 99.7%

Time: 15.3s

Alternatives: 13

Speedup: 1.7×

• 21.3% 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: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ x + \mathsf{fma}\left(z \cdot \left(x - y\right), 6, \left(y - x\right) \cdot 4\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (fma (* z (- x y)) 6.0 (* (- y x) 4.0))))

C

double code(double x, double y, double z) {
	return x + fma((z * (x - y)), 6.0, ((y - x) * 4.0));
}

Julia

function code(x, y, z)
	return Float64(x + fma(Float64(z * Float64(x - y)), 6.0, Float64(Float64(y - x) * 4.0)))
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision] * 6.0 + N[(N[(y - x), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $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. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*r* N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. neg-lowering-neg.f64 N/A

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

   8. --lowering--.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-lowering-*.f64 N/A

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

   12. --lowering--.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+135}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, -6, 4\right)\\ \mathbf{elif}\;t\_0 \leq 0.6666666666666665:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, 6, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (<= t_0 -1e+135)
     (* y (fma z -6.0 4.0))
     (if (<= t_0 0.6666666666666665)
       (* x (fma 6.0 z -3.0))
       (if (<= t_0 1.0) (fma 4.0 (- y x) x) (fma (* x z) 6.0 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if (t_0 <= -1e+135) {
		tmp = y * fma(z, -6.0, 4.0);
	} else if (t_0 <= 0.6666666666666665) {
		tmp = x * fma(6.0, z, -3.0);
	} else if (t_0 <= 1.0) {
		tmp = fma(4.0, (y - x), x);
	} else {
		tmp = fma((x * z), 6.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_0 <= -1e+135)
		tmp = Float64(y * fma(z, -6.0, 4.0));
	elseif (t_0 <= 0.6666666666666665)
		tmp = Float64(x * fma(6.0, z, -3.0));
	elseif (t_0 <= 1.0)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = fma(Float64(x * z), 6.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+135], N[(y * N[(z * -6.0 + 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.6666666666666665], 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], N[(N[(x * z), $MachinePrecision] * 6.0 + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -9.99999999999999962e134

   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 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. *-lowering-*.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. accelerator-lowering-fma.f64 75.9

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

   5. Simplified 75.9%

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

   if -9.99999999999999962e134 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.66666666666666652

   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 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. associate-*r* N/A

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

      13. mul-1-neg N/A

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

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

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

      15. *-lowering-*.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

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

      21. metadata-eval N/A

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

   5. Simplified 67.4%

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

   if 0.66666666666666652 < (-.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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 99.3

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

   5. Simplified 99.3%

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

   if 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. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\left(0 - \left(\frac{2}{3} - z\right)\right)}, 6, x\right) \]

      7. associate--r- N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\left(\left(0 - \frac{2}{3}\right) + z\right)}, 6, x\right) \]

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 62.0

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

   7. Simplified 62.0%

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

   8. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 62.0

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

   10. Simplified 62.0%

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

Alternative 3: 75.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(6, z, -3\right)\\ t_1 := \frac{2}{3} - z\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+135}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, -6, 4\right)\\ \mathbf{elif}\;t\_1 \leq 0.6666666666666665:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (fma 6.0 z -3.0))) (t_1 (- (/ 2.0 3.0) z)))
   (if (<= t_1 -1e+135)
     (* y (fma z -6.0 4.0))
     (if (<= t_1 0.6666666666666665)
       t_0
       (if (<= t_1 1.0) (fma 4.0 (- y x) x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * fma(6.0, z, -3.0);
	double t_1 = (2.0 / 3.0) - z;
	double tmp;
	if (t_1 <= -1e+135) {
		tmp = y * fma(z, -6.0, 4.0);
	} else if (t_1 <= 0.6666666666666665) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = fma(4.0, (y - x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * fma(6.0, z, -3.0))
	t_1 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_1 <= -1e+135)
		tmp = Float64(y * fma(z, -6.0, 4.0));
	elseif (t_1 <= 0.6666666666666665)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(6.0 * z + -3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+135], N[(y * N[(z * -6.0 + 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.6666666666666665], t$95$0, If[LessEqual[t$95$1, 1.0], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -9.99999999999999962e134

   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 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. *-lowering-*.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. accelerator-lowering-fma.f64 75.9

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

   5. Simplified 75.9%

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

   if -9.99999999999999962e134 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.66666666666666652 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. 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. associate-*r* N/A

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

      13. mul-1-neg N/A

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

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

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

      15. *-lowering-*.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

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

      21. metadata-eval N/A

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

   5. Simplified 63.4%

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

   if 0.66666666666666652 < (-.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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 99.3

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

   5. Simplified 99.3%

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

Alternative 4: 75.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(6, z, -3\right)\\ t_1 := \frac{2}{3} - z\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+135}:\\ \;\;\;\;y \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;t\_1 \leq 0.6666666666666665:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (fma 6.0 z -3.0))) (t_1 (- (/ 2.0 3.0) z)))
   (if (<= t_1 -1e+135)
     (* y (* z -6.0))
     (if (<= t_1 0.6666666666666665)
       t_0
       (if (<= t_1 1.0) (fma 4.0 (- y x) x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * fma(6.0, z, -3.0);
	double t_1 = (2.0 / 3.0) - z;
	double tmp;
	if (t_1 <= -1e+135) {
		tmp = y * (z * -6.0);
	} else if (t_1 <= 0.6666666666666665) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = fma(4.0, (y - x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * fma(6.0, z, -3.0))
	t_1 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_1 <= -1e+135)
		tmp = Float64(y * Float64(z * -6.0));
	elseif (t_1 <= 0.6666666666666665)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(6.0 * z + -3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+135], N[(y * N[(z * -6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.6666666666666665], t$95$0, If[LessEqual[t$95$1, 1.0], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -9.99999999999999962e134

   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 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. *-lowering-*.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. accelerator-lowering-fma.f64 75.9

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

   5. Simplified 75.9%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 75.9

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

   8. Simplified 75.9%

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

   if -9.99999999999999962e134 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.66666666666666652 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. 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. associate-*r* N/A

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

      13. mul-1-neg N/A

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

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

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

      15. *-lowering-*.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

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

      21. metadata-eval N/A

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

   5. Simplified 63.4%

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

   if 0.66666666666666652 < (-.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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 99.3

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

   5. Simplified 99.3%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -1 \cdot 10^{+135}:\\ \;\;\;\;y \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 0.6666666666666665:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(z \cdot 6\right)\\ t_1 := \frac{2}{3} - z\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+135}:\\ \;\;\;\;y \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* z 6.0))) (t_1 (- (/ 2.0 3.0) z)))
   (if (<= t_1 -1e+135)
     (* y (* z -6.0))
     (if (<= t_1 -1.0) t_0 (if (<= t_1 1.0) (fma 4.0 (- y x) x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * (z * 6.0);
	double t_1 = (2.0 / 3.0) - z;
	double tmp;
	if (t_1 <= -1e+135) {
		tmp = y * (z * -6.0);
	} else if (t_1 <= -1.0) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = fma(4.0, (y - x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(z * 6.0))
	t_1 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_1 <= -1e+135)
		tmp = Float64(y * Float64(z * -6.0));
	elseif (t_1 <= -1.0)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(z * 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+135], N[(y * N[(z * -6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1.0], t$95$0, If[LessEqual[t$95$1, 1.0], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -9.99999999999999962e134

   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 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. *-lowering-*.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. accelerator-lowering-fma.f64 75.9

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

   5. Simplified 75.9%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 75.9

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

   8. Simplified 75.9%

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

   if -9.99999999999999962e134 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1 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. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. 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 egg-rr 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 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\left(0 - \left(\frac{2}{3} - z\right)\right)}, 6, x\right) \]

      7. associate--r- N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\left(\left(0 - \frac{2}{3}\right) + z\right)}, 6, x\right) \]

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 63.0

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

   7. Simplified 63.0%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 61.1

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

   10. Simplified 61.1%

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

   if -1 < (-.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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 99.3

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

   5. Simplified 99.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -1 \cdot 10^{+135}:\\ \;\;\;\;y \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq -1:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

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

   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. *-lowering-*.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. *-lowering-*.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. --lowering--.f64 96.6

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

   5. Simplified 96.6%

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

   if -1 < (-.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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 99.3

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

   5. Simplified 99.3%

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

   if 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. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. --lowering--.f64 99.8

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

   7. Simplified 99.8%

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

4. Final simplification 98.8%

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

Alternative 7: 97.7% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (* z (- x y)) 6.0)))
   (if (<= t_0 -1.0) 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 = (z * (x - y)) * 6.0;
	double tmp;
	if (t_0 <= -1.0) {
		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(Float64(z * Float64(x - y)) * 6.0)
	tmp = 0.0
	if (t_0 <= -1.0)
		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[(N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], 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) < -1 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. 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. *-lowering-*.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. *-lowering-*.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. --lowering--.f64 98.3

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

   5. Simplified 98.3%

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

   if -1 < (-.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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 99.3

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

   5. Simplified 99.3%

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

4. Final simplification 98.8%

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

Alternative 8: 74.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(z \cdot 6\right)\\ \mathbf{if}\;z \leq -180:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* z 6.0))))
   (if (<= z -180.0) t_0 (if (<= z 0.5) (fma 4.0 (- y x) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (z * 6.0);
	double tmp;
	if (z <= -180.0) {
		tmp = t_0;
	} else if (z <= 0.5) {
		tmp = fma(4.0, (y - x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(z * 6.0))
	tmp = 0.0
	if (z <= -180.0)
		tmp = t_0;
	elseif (z <= 0.5)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(z * 6.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -180.0], t$95$0, If[LessEqual[z, 0.5], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -180 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. 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 egg-rr 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 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\left(0 - \left(\frac{2}{3} - z\right)\right)}, 6, x\right) \]

      7. associate--r- N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\left(\left(0 - \frac{2}{3}\right) + z\right)}, 6, x\right) \]

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 57.2

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

   7. Simplified 57.2%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 55.8

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

   10. Simplified 55.8%

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

   if -180 < 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. 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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 99.3

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

   5. Simplified 99.3%

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

Alternative 9: 35.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-180}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+80}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6e-180) (* y 4.0) (if (<= y 5.8e+80) (* x -3.0) (fma 4.0 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6e-180) {
		tmp = y * 4.0;
	} else if (y <= 5.8e+80) {
		tmp = x * -3.0;
	} else {
		tmp = fma(4.0, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6e-180)
		tmp = Float64(y * 4.0);
	elseif (y <= 5.8e+80)
		tmp = Float64(x * -3.0);
	else
		tmp = fma(4.0, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6e-180], N[(y * 4.0), $MachinePrecision], If[LessEqual[y, 5.8e+80], N[(x * -3.0), $MachinePrecision], N[(4.0 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.0000000000000001e-180

   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. 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. *-lowering-*.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. accelerator-lowering-fma.f64 68.3

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

   5. Simplified 68.3%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 40.6%

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

   if -6.0000000000000001e-180 < y < 5.79999999999999971e80

   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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 45.2

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

   5. Simplified 45.2%

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

   6. Taylor expanded in y around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 34.1

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

   8. Simplified 34.1%

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

   if 5.79999999999999971e80 < 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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 58.4

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

   5. Simplified 58.4%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(4, \color{blue}{y}, x\right) \]
   7. Step-by-step derivation

   8. Simplified 48.7%

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

Alternative 10: 35.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-180}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{+72}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6e-180) (* y 4.0) (if (<= y 1.66e+72) (* x -3.0) (* y 4.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6e-180) {
		tmp = y * 4.0;
	} else if (y <= 1.66e+72) {
		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 <= (-6d-180)) then
        tmp = y * 4.0d0
    else if (y <= 1.66d+72) 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 <= -6e-180) {
		tmp = y * 4.0;
	} else if (y <= 1.66e+72) {
		tmp = x * -3.0;
	} else {
		tmp = y * 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6e-180:
		tmp = y * 4.0
	elif y <= 1.66e+72:
		tmp = x * -3.0
	else:
		tmp = y * 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6e-180)
		tmp = Float64(y * 4.0);
	elseif (y <= 1.66e+72)
		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 <= -6e-180)
		tmp = y * 4.0;
	elseif (y <= 1.66e+72)
		tmp = x * -3.0;
	else
		tmp = y * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6e-180], N[(y * 4.0), $MachinePrecision], If[LessEqual[y, 1.66e+72], N[(x * -3.0), $MachinePrecision], N[(y * 4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.0000000000000001e-180 or 1.6599999999999999e72 < y

   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. 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. *-lowering-*.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. accelerator-lowering-fma.f64 74.3

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

   5. Simplified 74.3%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 43.1%

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

   if -6.0000000000000001e-180 < y < 1.6599999999999999e72

   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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 44.6

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

   5. Simplified 44.6%

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

   6. Taylor expanded in y around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 34.1

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

   8. Simplified 34.1%

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

Alternative 11: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. --lowering--.f64 N/A

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

   9. 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 egg-rr 99.6%

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

Alternative 12: 50.4% 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. accelerator-lowering-fma.f64 N/A

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

   3. --lowering--.f64 50.8

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

5. Simplified 50.8%

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

Alternative 13: 25.5% 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. accelerator-lowering-fma.f64 N/A

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

   3. --lowering--.f64 50.8

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

5. Simplified 50.8%

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

6. Taylor expanded in y around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 21.5

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

8. Simplified 21.5%

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

Reproduce

herbie shell --seed 2024198 
(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))))