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

Percentage Accurate: 99.5% → 99.8%

Time: 13.3s

Alternatives: 13

Speedup: 1.9×

• 21.1% 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: 99.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

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

5. Simplified 99.8%

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

Alternative 2: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(z \cdot -6\right)\\ t_1 := \frac{2}{3} - z\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+116}:\\ \;\;\;\;6 \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq -200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+226}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* z -6.0))) (t_1 (- (/ 2.0 3.0) z)))
   (if (<= t_1 -5e+116)
     (* 6.0 (* z x))
     (if (<= t_1 -200000.0)
       t_0
       (if (<= t_1 1.0)
         (fma 4.0 (- y x) x)
         (if (<= t_1 1e+226) (* x (fma 6.0 z -3.0)) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (z * -6.0);
	double t_1 = (2.0 / 3.0) - z;
	double tmp;
	if (t_1 <= -5e+116) {
		tmp = 6.0 * (z * x);
	} else if (t_1 <= -200000.0) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = fma(4.0, (y - x), x);
	} else if (t_1 <= 1e+226) {
		tmp = x * fma(6.0, z, -3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(z * -6.0))
	t_1 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_1 <= -5e+116)
		tmp = Float64(6.0 * Float64(z * x));
	elseif (t_1 <= -200000.0)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = fma(4.0, Float64(y - x), x);
	elseif (t_1 <= 1e+226)
		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), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+116], N[(6.0 * N[(z * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -200000.0], t$95$0, If[LessEqual[t$95$1, 1.0], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+226], N[(x * N[(6.0 * z + -3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -5.00000000000000025e116

   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 y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 65.6

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

   5. Simplified 65.6%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 65.7

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

   8. Simplified 65.7%

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

   if -5.00000000000000025e116 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -2e5 or 9.99999999999999961e225 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 65.3

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

   5. Simplified 65.3%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 64.1

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

   8. Simplified 64.1%

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

   if -2e5 < (-.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 97.4

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

   5. Simplified 97.4%

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

   if 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 9.99999999999999961e225

   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. 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

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

      21. metadata-eval N/A

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

   5. Simplified 57.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -5 \cdot 10^{+116}:\\ \;\;\;\;6 \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq -200000:\\ \;\;\;\;y \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 10^{+226}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot -6\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(z \cdot -6\right)\\ t_1 := \frac{2}{3} - z\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+116}:\\ \;\;\;\;6 \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq -200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+226}:\\ \;\;\;\;x \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* z -6.0))) (t_1 (- (/ 2.0 3.0) z)))
   (if (<= t_1 -5e+116)
     (* 6.0 (* z x))
     (if (<= t_1 -200000.0)
       t_0
       (if (<= t_1 2.0)
         (fma 4.0 (- y x) x)
         (if (<= t_1 1e+226) (* x (* 6.0 z)) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (z * -6.0);
	double t_1 = (2.0 / 3.0) - z;
	double tmp;
	if (t_1 <= -5e+116) {
		tmp = 6.0 * (z * x);
	} else if (t_1 <= -200000.0) {
		tmp = t_0;
	} else if (t_1 <= 2.0) {
		tmp = fma(4.0, (y - x), x);
	} else if (t_1 <= 1e+226) {
		tmp = x * (6.0 * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(z * -6.0))
	t_1 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_1 <= -5e+116)
		tmp = Float64(6.0 * Float64(z * x));
	elseif (t_1 <= -200000.0)
		tmp = t_0;
	elseif (t_1 <= 2.0)
		tmp = fma(4.0, Float64(y - x), x);
	elseif (t_1 <= 1e+226)
		tmp = Float64(x * Float64(6.0 * z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(z * -6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+116], N[(6.0 * N[(z * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -200000.0], t$95$0, If[LessEqual[t$95$1, 2.0], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+226], N[(x * N[(6.0 * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -5.00000000000000025e116

   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 y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 65.6

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

   5. Simplified 65.6%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 65.7

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

   8. Simplified 65.7%

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

   if -5.00000000000000025e116 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -2e5 or 9.99999999999999961e225 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 65.3

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

   5. Simplified 65.3%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 64.1

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

   8. Simplified 64.1%

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

   if -2e5 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 2

   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 96.6

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

   5. Simplified 96.6%

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

   if 2 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 9.99999999999999961e225

   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. 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

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

      21. metadata-eval N/A

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

   5. Simplified 56.3%

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

   6. Taylor expanded in z around inf

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

      1. metadata-eval N/A

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

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

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-lft-neg-out N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 54.7

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

   8. Simplified 54.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -5 \cdot 10^{+116}:\\ \;\;\;\;6 \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq -200000:\\ \;\;\;\;y \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 2:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 10^{+226}:\\ \;\;\;\;x \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot -6\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := \left(y - x\right) \cdot \left(z \cdot -6\right)\\ \mathbf{if}\;t\_0 \leq -200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (- y x) (* z -6.0))))
   (if (<= t_0 -200000.0) t_1 (if (<= t_0 2.0) (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 - x) * (z * -6.0);
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		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(y - x) * Float64(z * -6.0))
	tmp = 0.0
	if (t_0 <= -200000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		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[(y - x), $MachinePrecision] * N[(z * -6.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -200000.0], t$95$1, If[LessEqual[t$95$0, 2.0], 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) < -2e5 or 2 < (-.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

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

      7. lift-/.f64 N/A

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

      8. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.9

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

   7. Simplified 98.9%

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

   if -2e5 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 2

   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 96.6

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

   5. Simplified 96.6%

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

Alternative 5: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := 6 \cdot \left(z \cdot \left(x - y\right)\right)\\ \mathbf{if}\;t\_0 \leq -200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* 6.0 (* z (- x y)))))
   (if (<= t_0 -200000.0) t_1 (if (<= t_0 2.0) (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 = 6.0 * (z * (x - y));
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		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(6.0 * Float64(z * Float64(x - y)))
	tmp = 0.0
	if (t_0 <= -200000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		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[(6.0 * N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -200000.0], t$95$1, If[LessEqual[t$95$0, 2.0], 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) < -2e5 or 2 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. 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 98.8

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

   5. Simplified 98.8%

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

   if -2e5 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 2

   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 96.6

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

   5. Simplified 96.6%

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

Alternative 6: 62.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := -z \cdot y\\ \mathbf{if}\;t\_0 \leq -200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (- (* z y))))
   (if (<= t_0 -200000.0)
     t_1
     (if (<= t_0 100000000.0) (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 * y);
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = t_1;
	} else if (t_0 <= 100000000.0) {
		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 * y))
	tmp = 0.0
	if (t_0 <= -200000.0)
		tmp = t_1;
	elseif (t_0 <= 100000000.0)
		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[(z * y), $MachinePrecision])}, If[LessEqual[t$95$0, -200000.0], t$95$1, If[LessEqual[t$95$0, 100000000.0], 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) < -2e5 or 1e8 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 52.5

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

   5. Simplified 52.5%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 52.1

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

   8. Simplified 52.1%

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

   9. Taylor expanded in z around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 31.9

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

   11. Simplified 31.9%

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

   if -2e5 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1e8

   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 95.1

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

   5. Simplified 95.1%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -200000:\\ \;\;\;\;-z \cdot y\\ \mathbf{elif}\;\frac{2}{3} - z \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -z \cdot y\\ \mathbf{if}\;z \leq -35000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.96 \cdot 10^{-289}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 0.000465:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* z y))))
   (if (<= z -35000000000.0)
     t_0
     (if (<= z 1.96e-289) (* x -3.0) (if (<= z 0.000465) (* y 4.0) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = -(z * y);
	double tmp;
	if (z <= -35000000000.0) {
		tmp = t_0;
	} else if (z <= 1.96e-289) {
		tmp = x * -3.0;
	} else if (z <= 0.000465) {
		tmp = y * 4.0;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = -(z * y)
    if (z <= (-35000000000.0d0)) then
        tmp = t_0
    else if (z <= 1.96d-289) then
        tmp = x * (-3.0d0)
    else if (z <= 0.000465d0) then
        tmp = y * 4.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -(z * y);
	double tmp;
	if (z <= -35000000000.0) {
		tmp = t_0;
	} else if (z <= 1.96e-289) {
		tmp = x * -3.0;
	} else if (z <= 0.000465) {
		tmp = y * 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(z * y)
	tmp = 0
	if z <= -35000000000.0:
		tmp = t_0
	elif z <= 1.96e-289:
		tmp = x * -3.0
	elif z <= 0.000465:
		tmp = y * 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(z * y))
	tmp = 0.0
	if (z <= -35000000000.0)
		tmp = t_0;
	elseif (z <= 1.96e-289)
		tmp = Float64(x * -3.0);
	elseif (z <= 0.000465)
		tmp = Float64(y * 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(z * y);
	tmp = 0.0;
	if (z <= -35000000000.0)
		tmp = t_0;
	elseif (z <= 1.96e-289)
		tmp = x * -3.0;
	elseif (z <= 0.000465)
		tmp = y * 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(z * y), $MachinePrecision])}, If[LessEqual[z, -35000000000.0], t$95$0, If[LessEqual[z, 1.96e-289], N[(x * -3.0), $MachinePrecision], If[LessEqual[z, 0.000465], N[(y * 4.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5e10 or 4.6500000000000003e-4 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 52.2

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

   5. Simplified 52.2%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.8

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

   8. Simplified 51.8%

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

   9. Taylor expanded in z around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 31.7

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

   11. Simplified 31.7%

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

   if -3.5e10 < z < 1.96e-289

   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 93.9

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

   5. Simplified 93.9%

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

   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. lower-*.f64 54.2

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

   8. Simplified 54.2%

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

   if 1.96e-289 < z < 4.6500000000000003e-4

   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 97.8

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

   5. Simplified 97.8%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 56.5

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

   8. Simplified 56.5%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -35000000000:\\ \;\;\;\;-z \cdot y\\ \mathbf{elif}\;z \leq 1.96 \cdot 10^{-289}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 0.000465:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \mathsf{fma}\left(z, -6, 4\right)\\ \mathbf{if}\;y \leq -116000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (fma z -6.0 4.0))))
   (if (<= y -116000.0) t_0 (if (<= y 2.5e-51) (* 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 <= -116000.0) {
		tmp = t_0;
	} else if (y <= 2.5e-51) {
		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 <= -116000.0)
		tmp = t_0;
	elseif (y <= 2.5e-51)
		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, -116000.0], t$95$0, If[LessEqual[y, 2.5e-51], N[(x * N[(6.0 * z + -3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -116000 or 2.50000000000000002e-51 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. 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 81.9

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

   5. Simplified 81.9%

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

   if -116000 < y < 2.50000000000000002e-51

   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 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. 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

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

      21. metadata-eval N/A

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

   5. Simplified 85.6%

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

Alternative 9: 74.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -14.6:\\ \;\;\;\;x \cdot \left(6 \cdot z\right)\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(z \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -14.6)
   (* x (* 6.0 z))
   (if (<= z 0.5) (fma 4.0 (- y x) x) (* 6.0 (* z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -14.6) {
		tmp = x * (6.0 * z);
	} else if (z <= 0.5) {
		tmp = fma(4.0, (y - x), x);
	} else {
		tmp = 6.0 * (z * x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -14.6)
		tmp = Float64(x * Float64(6.0 * z));
	elseif (z <= 0.5)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = Float64(6.0 * Float64(z * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -14.6], N[(x * N[(6.0 * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.5], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(6.0 * N[(z * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -14.5999999999999996

   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 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. 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

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

      21. metadata-eval N/A

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

   5. Simplified 53.6%

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

   6. Taylor expanded in z around inf

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

      1. metadata-eval N/A

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

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

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-lft-neg-out N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 52.4

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

   8. Simplified 52.4%

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

   if -14.5999999999999996 < 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. lower-fma.f64 N/A

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

      3. lower--.f64 96.6

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

   5. Simplified 96.6%

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

   if 0.5 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 56.0

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

   5. Simplified 56.0%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.1

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

   8. Simplified 56.1%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -14.6:\\ \;\;\;\;x \cdot \left(6 \cdot z\right)\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(z \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(z \cdot x\right)\\ \mathbf{if}\;z \leq -14.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 6.0 (* z x))))
   (if (<= z -14.6) 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 = 6.0 * (z * x);
	double tmp;
	if (z <= -14.6) {
		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(6.0 * Float64(z * x))
	tmp = 0.0
	if (z <= -14.6)
		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[(6.0 * N[(z * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -14.6], 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 < -14.5999999999999996 or 0.5 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 54.8

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

   5. Simplified 54.8%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 54.2

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

   8. Simplified 54.2%

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

   if -14.5999999999999996 < 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. lower-fma.f64 N/A

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

      3. lower--.f64 96.6

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

   5. Simplified 96.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -14.6:\\ \;\;\;\;6 \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(z \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -30000000000:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-54}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -30000000000.0) (* y 4.0) (if (<= y 1.5e-54) (* x -3.0) (* y 4.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -30000000000.0) {
		tmp = y * 4.0;
	} else if (y <= 1.5e-54) {
		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 <= (-30000000000.0d0)) then
        tmp = y * 4.0d0
    else if (y <= 1.5d-54) 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 <= -30000000000.0) {
		tmp = y * 4.0;
	} else if (y <= 1.5e-54) {
		tmp = x * -3.0;
	} else {
		tmp = y * 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -30000000000.0:
		tmp = y * 4.0
	elif y <= 1.5e-54:
		tmp = x * -3.0
	else:
		tmp = y * 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -30000000000.0)
		tmp = Float64(y * 4.0);
	elseif (y <= 1.5e-54)
		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 <= -30000000000.0)
		tmp = y * 4.0;
	elseif (y <= 1.5e-54)
		tmp = x * -3.0;
	else
		tmp = y * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -30000000000.0], N[(y * 4.0), $MachinePrecision], If[LessEqual[y, 1.5e-54], N[(x * -3.0), $MachinePrecision], N[(y * 4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3e10 or 1.50000000000000005e-54 < y

   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 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 40.9

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

   5. Simplified 40.9%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 35.1

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

   8. Simplified 35.1%

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

   if -3e10 < y < 1.50000000000000005e-54

   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 46.4

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

   5. Simplified 46.4%

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

   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. lower-*.f64 40.1

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

   8. Simplified 40.1%

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

Alternative 12: 25.3% 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 99.6%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
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 43.5

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

5. Simplified 43.5%

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

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. lower-*.f64 23.4

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

8. Simplified 23.4%

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

Alternative 13: 6.6% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-x)

Derivation

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 43.5

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

5. Simplified 43.5%

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

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. lower-*.f64 23.4

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

8. Simplified 23.4%

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

9. Taylor expanded in x around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 6.5

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

11. Simplified 6.5%

    \[\leadsto \color{blue}{-x} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))