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

Percentage Accurate: 99.5% → 99.6%

Time: 16.2s

Alternatives: 17

Speedup: 1.7×

• 20.7% 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: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{\mathsf{fma}\left(y - x, \mathsf{fma}\left(-6, z, 4\right), x\right)}} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ 1.0 (/ 1.0 (fma (- y x) (fma -6.0 z 4.0) x))))

C

double code(double x, double y, double z) {
	return 1.0 / (1.0 / fma((y - x), fma(-6.0, z, 4.0), x));
}

Julia

function code(x, y, z)
	return Float64(1.0 / Float64(1.0 / fma(Float64(y - x), fma(-6.0, z, 4.0), x)))
end

Wolfram

code[x_, y_, z_] := N[(1.0 / N[(1.0 / N[(N[(y - x), $MachinePrecision] * N[(-6.0 * z + 4.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. flip3-+ N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. flip3-+ N/A

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(y - x, \mathsf{fma}\left(-6, z, 4\right), x\right)}}} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -20000000:\\ \;\;\;\;-6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, -3, y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(x - y\right), 6, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (<= t_0 -20000000.0)
     (* -6.0 (* (- y x) z))
     (if (<= t_0 1.0) (fma x -3.0 (* y 4.0)) (fma (* z (- x y)) 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 <= -20000000.0) {
		tmp = -6.0 * ((y - x) * z);
	} else if (t_0 <= 1.0) {
		tmp = fma(x, -3.0, (y * 4.0));
	} else {
		tmp = fma((z * (x - y)), 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 <= -20000000.0)
		tmp = Float64(-6.0 * Float64(Float64(y - x) * z));
	elseif (t_0 <= 1.0)
		tmp = fma(x, -3.0, Float64(y * 4.0));
	else
		tmp = fma(Float64(z * Float64(x - y)), 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, -20000000.0], N[(-6.0 * N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(x * -3.0 + N[(y * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision] * 6.0 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -2e7

   1. Initial program 99.6%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(y - x, \mathsf{fma}\left(-6, z, 4\right), x\right)}}} \]

   5. Taylor expanded in z around inf

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

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

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

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

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

      3. --lowering--.f64 98.9

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

   7. Simplified 98.9%

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

   if -2e7 < (-.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 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 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 97.8

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

   8. Simplified 97.8%

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

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

   1. Initial program 99.6%

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

      1. +-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 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. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

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

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

      11. --lowering--.f64 98.2

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

   7. Simplified 98.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -20000000:\\ \;\;\;\;-6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, -3, y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(x - y\right), 6, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -20000000:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (<= t_0 -20000000.0)
     (* 6.0 (* x z))
     (if (<= t_0 1.0) (fma 4.0 (- y x) x) (* x (* z 6.0))))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if (t_0 <= -20000000.0) {
		tmp = 6.0 * (x * z);
	} else if (t_0 <= 1.0) {
		tmp = fma(4.0, (y - x), x);
	} else {
		tmp = x * (z * 6.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_0 <= -20000000.0)
		tmp = Float64(6.0 * Float64(x * z));
	elseif (t_0 <= 1.0)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = Float64(x * Float64(z * 6.0));
	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, -20000000.0], N[(6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(z * 6.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -2e7

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. --lowering--.f64 98.8

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 62.1

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

   8. Simplified 62.1%

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

   if -2e7 < (-.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 97.8

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

   5. Simplified 97.8%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. --lowering--.f64 98.1

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

   5. Simplified 98.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

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

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lowering-*.f64 98.1

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

   7. Applied egg-rr 98.1%

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

   8. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. *-lowering-*.f64 50.7

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

   10. Simplified 50.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -20000000:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -20000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* 6.0 (* x z))))
   (if (<= t_0 -20000000.0) 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 = 6.0 * (x * z);
	double tmp;
	if (t_0 <= -20000000.0) {
		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(6.0 * Float64(x * z))
	tmp = 0.0
	if (t_0 <= -20000000.0)
		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[(6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20000000.0], 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) < -2e7 or 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. --lowering--.f64 98.4

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

   5. Simplified 98.4%

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

   6. Taylor expanded in x around inf

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 56.3

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

   8. Simplified 56.3%

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

   if -2e7 < (-.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 97.8

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

   5. Simplified 97.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -20000000:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ x + \frac{\mathsf{fma}\left(-6, z, 4\right)}{\frac{1}{y - x}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (/ (fma -6.0 z 4.0) (/ 1.0 (- y x)))))

C

double code(double x, double y, double z) {
	return x + (fma(-6.0, z, 4.0) / (1.0 / (y - x)));
}

Julia

function code(x, y, z)
	return Float64(x + Float64(fma(-6.0, z, 4.0) / Float64(1.0 / Float64(y - x))))
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(-6.0 * z + 4.0), $MachinePrecision] / N[(1.0 / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

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

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. distribute-lft-in N/A

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

   10. neg-mul-1 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. clear-num N/A

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

   19. flip-- N/A

       \[\leadsto x + \frac{\mathsf{fma}\left(-6, z, 4\right)}{\frac{1}{\color{blue}{y - x}}} \]

   20. /-lowering-/.f64 N/A

       \[\leadsto x + \frac{\mathsf{fma}\left(-6, z, 4\right)}{\color{blue}{\frac{1}{y - x}}} \]

   21. --lowering--.f64 99.7

       \[\leadsto x + \frac{\mathsf{fma}\left(-6, z, 4\right)}{\frac{1}{\color{blue}{y - x}}} \]

4. Applied egg-rr 99.7%

   \[\leadsto x + \color{blue}{\frac{\mathsf{fma}\left(-6, z, 4\right)}{\frac{1}{y - x}}} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \mathbf{if}\;z \leq -9 \cdot 10^{+71}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(x, -3, y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (fma 6.0 z -3.0))))
   (if (<= z -9e+71)
     (* -6.0 (* y z))
     (if (<= z -2.9e-9) t_0 (if (<= z 1.1e-27) (fma x -3.0 (* y 4.0)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * fma(6.0, z, -3.0);
	double tmp;
	if (z <= -9e+71) {
		tmp = -6.0 * (y * z);
	} else if (z <= -2.9e-9) {
		tmp = t_0;
	} else if (z <= 1.1e-27) {
		tmp = fma(x, -3.0, (y * 4.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * fma(6.0, z, -3.0))
	tmp = 0.0
	if (z <= -9e+71)
		tmp = Float64(-6.0 * Float64(y * z));
	elseif (z <= -2.9e-9)
		tmp = t_0;
	elseif (z <= 1.1e-27)
		tmp = fma(x, -3.0, Float64(y * 4.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(6.0 * z + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9e+71], N[(-6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2.9e-9], t$95$0, If[LessEqual[z, 1.1e-27], N[(x * -3.0 + N[(y * 4.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.00000000000000087e71

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. --lowering--.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 63.7

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

   8. Simplified 63.7%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 63.8

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

   10. Applied egg-rr 63.8%

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

   if -9.00000000000000087e71 < z < -2.89999999999999991e-9 or 1.09999999999999993e-27 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval 66.8

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

   5. Simplified 66.8%

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

   if -2.89999999999999991e-9 < z < 1.09999999999999993e-27

   1. Initial program 99.2%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   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 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 99.5

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

   8. Simplified 99.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+71}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(x, -3, y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \mathbf{if}\;z \leq -8 \cdot 10^{+71}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (fma 6.0 z -3.0))))
   (if (<= z -8e+71)
     (* -6.0 (* y z))
     (if (<= z -5e-8) t_0 (if (<= z 1.35e-28) (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 tmp;
	if (z <= -8e+71) {
		tmp = -6.0 * (y * z);
	} else if (z <= -5e-8) {
		tmp = t_0;
	} else if (z <= 1.35e-28) {
		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))
	tmp = 0.0
	if (z <= -8e+71)
		tmp = Float64(-6.0 * Float64(y * z));
	elseif (z <= -5e-8)
		tmp = t_0;
	elseif (z <= 1.35e-28)
		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]}, If[LessEqual[z, -8e+71], N[(-6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -5e-8], t$95$0, If[LessEqual[z, 1.35e-28], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.0000000000000003e71

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. --lowering--.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 63.7

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

   8. Simplified 63.7%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 63.8

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

   10. Applied egg-rr 63.8%

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

   if -8.0000000000000003e71 < z < -4.9999999999999998e-8 or 1.3499999999999999e-28 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval 67.6

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

   5. Simplified 67.6%

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

   if -4.9999999999999998e-8 < z < 1.3499999999999999e-28

   1. Initial program 99.2%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   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 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+71}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+72}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -14.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 6.0 (* x z))))
   (if (<= z -1e+72)
     (* -6.0 (* y z))
     (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 * (x * z);
	double tmp;
	if (z <= -1e+72) {
		tmp = -6.0 * (y * z);
	} else 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(x * z))
	tmp = 0.0
	if (z <= -1e+72)
		tmp = Float64(-6.0 * Float64(y * z));
	elseif (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[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+72], N[(-6.0 * N[(y * z), $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 3 regimes

2. if z < -9.99999999999999944e71

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. --lowering--.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 63.7

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

   8. Simplified 63.7%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 63.8

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

   10. Applied egg-rr 63.8%

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

   if -9.99999999999999944e71 < z < -14.5999999999999996 or 0.5 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. --lowering--.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around inf

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 64.5

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

   8. Simplified 64.5%

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

   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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 97.8

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

   5. Simplified 97.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+72}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -14.6:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+71}:\\ \;\;\;\;y \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;z \leq -3.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.58:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 6.0 (* x z))))
   (if (<= z -4.2e+71)
     (* y (* -6.0 z))
     (if (<= z -3.4) t_0 (if (<= z 0.58) (fma 4.0 (- y x) x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = 6.0 * (x * z);
	double tmp;
	if (z <= -4.2e+71) {
		tmp = y * (-6.0 * z);
	} else if (z <= -3.4) {
		tmp = t_0;
	} else if (z <= 0.58) {
		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(x * z))
	tmp = 0.0
	if (z <= -4.2e+71)
		tmp = Float64(y * Float64(-6.0 * z));
	elseif (z <= -3.4)
		tmp = t_0;
	elseif (z <= 0.58)
		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[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.2e+71], N[(y * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3.4], t$95$0, If[LessEqual[z, 0.58], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.19999999999999978e71

   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. flip3-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(y - x, \mathsf{fma}\left(-6, z, 4\right), x\right)}}} \]

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 63.7

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

   7. Simplified 63.7%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 63.8

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

   10. Simplified 63.8%

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

   if -4.19999999999999978e71 < z < -3.39999999999999991 or 0.57999999999999996 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. --lowering--.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around inf

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 64.5

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

   8. Simplified 64.5%

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

   if -3.39999999999999991 < z < 0.57999999999999996

   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 97.8

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

   5. Simplified 97.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+71}:\\ \;\;\;\;y \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;z \leq -3.4:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq 0.58:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+71}:\\ \;\;\;\;z \cdot \left(y \cdot -6\right)\\ \mathbf{elif}\;z \leq -232:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 6.0 (* x z))))
   (if (<= z -9.5e+71)
     (* z (* y -6.0))
     (if (<= z -232.0) 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 = 6.0 * (x * z);
	double tmp;
	if (z <= -9.5e+71) {
		tmp = z * (y * -6.0);
	} else if (z <= -232.0) {
		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(6.0 * Float64(x * z))
	tmp = 0.0
	if (z <= -9.5e+71)
		tmp = Float64(z * Float64(y * -6.0));
	elseif (z <= -232.0)
		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[(6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.5e+71], N[(z * N[(y * -6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -232.0], 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 3 regimes

2. if z < -9.50000000000000015e71

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. --lowering--.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 63.7

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

   8. Simplified 63.7%

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

   if -9.50000000000000015e71 < z < -232 or 0.5 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. --lowering--.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around inf

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 64.5

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

   8. Simplified 64.5%

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

   if -232 < 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 97.8

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

   5. Simplified 97.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+71}:\\ \;\;\;\;z \cdot \left(y \cdot -6\right)\\ \mathbf{elif}\;z \leq -232:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -0.57999999999999996 or 0.5 < z

   1. Initial program 99.6%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(y - x, \mathsf{fma}\left(-6, z, 4\right), x\right)}}} \]

   5. Taylor expanded in z around inf

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

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

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

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

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

      3. --lowering--.f64 98.5

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

   7. Simplified 98.5%

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

   if -0.57999999999999996 < z < 0.5

   1. Initial program 99.3%

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

   3. 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 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 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 97.8

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

   8. Simplified 97.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.58:\\ \;\;\;\;-6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(x, -3, y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

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 + (((y - x) * (0.6666666666666666d0 - z)) * 6.0d0)
end function

Java

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

Python

def code(x, y, z):
	return x + (((y - x) * (0.6666666666666666 - z)) * 6.0)

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * (0.6666666666666666 - z)) * 6.0);
end

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   8. metadata-eval 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 13: 36.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-83}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+157}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.2e-83) (* y 4.0) (if (<= y 1.35e+157) (* x -3.0) (fma 4.0 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e-83) {
		tmp = y * 4.0;
	} else if (y <= 1.35e+157) {
		tmp = x * -3.0;
	} else {
		tmp = fma(4.0, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.2e-83)
		tmp = Float64(y * 4.0);
	elseif (y <= 1.35e+157)
		tmp = Float64(x * -3.0);
	else
		tmp = fma(4.0, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.2e-83], N[(y * 4.0), $MachinePrecision], If[LessEqual[y, 1.35e+157], N[(x * -3.0), $MachinePrecision], N[(4.0 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.20000000000000025e-83

   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 57.7

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

   5. Simplified 57.7%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 47.1

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

   8. Simplified 47.1%

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

   if -7.20000000000000025e-83 < y < 1.35e157

   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 51.8

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

   5. Simplified 51.8%

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

   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 44.9

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

   8. Simplified 44.9%

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

   if 1.35e157 < 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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 49.0

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

   5. Simplified 49.0%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 43.5%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-83}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+157}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-83}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+157}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.2e-83) (* y 4.0) (if (<= y 1.25e+157) (* x -3.0) (* y 4.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.2e-83) {
		tmp = y * 4.0;
	} else if (y <= 1.25e+157) {
		tmp = x * -3.0;
	} else {
		tmp = y * 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.2d-83)) then
        tmp = y * 4.0d0
    else if (y <= 1.25d+157) then
        tmp = x * (-3.0d0)
    else
        tmp = y * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.2e-83) {
		tmp = y * 4.0;
	} else if (y <= 1.25e+157) {
		tmp = x * -3.0;
	} else {
		tmp = y * 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.2e-83:
		tmp = y * 4.0
	elif y <= 1.25e+157:
		tmp = x * -3.0
	else:
		tmp = y * 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.2e-83)
		tmp = Float64(y * 4.0);
	elseif (y <= 1.25e+157)
		tmp = Float64(x * -3.0);
	else
		tmp = Float64(y * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.2e-83)
		tmp = y * 4.0;
	elseif (y <= 1.25e+157)
		tmp = x * -3.0;
	else
		tmp = y * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.2e-83], N[(y * 4.0), $MachinePrecision], If[LessEqual[y, 1.25e+157], N[(x * -3.0), $MachinePrecision], N[(y * 4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1999999999999998e-83 or 1.24999999999999994e157 < 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 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 55.3

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

   5. Simplified 55.3%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 46.0

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

   8. Simplified 46.0%

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

   if -4.1999999999999998e-83 < y < 1.24999999999999994e157

   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 51.8

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

   5. Simplified 51.8%

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

   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 44.9

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

   8. Simplified 44.9%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-83}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+157}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 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 99.4%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
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 99.5

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

4. Applied egg-rr 99.5%

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

Alternative 16: 51.2% 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 99.4%

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

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

   3. --lowering--.f64 53.4

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

5. Simplified 53.4%

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

Alternative 17: 26.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ y \cdot 4 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* y 4.0))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y * 4.0d0
end function

Java

public static double code(double x, double y, double z) {
	return y * 4.0;
}

Python

def code(x, y, z):
	return y * 4.0

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = y * 4.0;
end

Wolfram

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

Derivation

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. accelerator-lowering-fma.f64 N/A

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

   3. --lowering--.f64 53.4

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

5. Simplified 53.4%

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

6. Taylor expanded in y around inf

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

   1. *-lowering-*.f64 25.7

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

8. Simplified 25.7%

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

9. Final simplification 25.7%

   \[\leadsto y \cdot 4 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(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))))