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

Percentage Accurate: 99.7% → 99.7%

Time: 10.3s

Alternatives: 13

Speedup: 1.1×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * 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) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * 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) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x + \frac{6 \cdot z}{\frac{1}{y - x}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((6.0d0 * z) / (1.0d0 / (y - x)))
end function

Java

public static double code(double x, double y, double z) {
	return x + ((6.0 * z) / (1.0 / (y - x)));
}

Python

def code(x, y, z):
	return x + ((6.0 * z) / (1.0 / (y - x)))

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((6.0 * z) / (1.0 / (y - x)));
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

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

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

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

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

   8. clear-num N/A

      \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{\frac{y \cdot y - x \cdot x}{y + x}}}} \]

   9. flip-- N/A

      \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

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

       \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{y - x}}} \]

   11. --lowering--.f64 99.8

       \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

4. Applied egg-rr 99.8%

   \[\leadsto x + \color{blue}{\frac{6 \cdot z}{\frac{1}{y - x}}} \]
5. Add Preprocessing

Alternative 2: 61.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(z \cdot y\right)\\ t_1 := \mathsf{fma}\left(-6, x \cdot z, 0\right)\\ \mathbf{if}\;z \leq -8 \cdot 10^{+71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -3250000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 27500000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 6.0 (* z y))) (t_1 (fma -6.0 (* x z) 0.0)))
   (if (<= z -8e+71)
     t_0
     (if (<= z -3250000000000.0)
       t_1
       (if (<= z -4.7e-57) t_0 (if (<= z 27500000.0) x t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = 6.0 * (z * y);
	double t_1 = fma(-6.0, (x * z), 0.0);
	double tmp;
	if (z <= -8e+71) {
		tmp = t_0;
	} else if (z <= -3250000000000.0) {
		tmp = t_1;
	} else if (z <= -4.7e-57) {
		tmp = t_0;
	} else if (z <= 27500000.0) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(6.0 * Float64(z * y))
	t_1 = fma(-6.0, Float64(x * z), 0.0)
	tmp = 0.0
	if (z <= -8e+71)
		tmp = t_0;
	elseif (z <= -3250000000000.0)
		tmp = t_1;
	elseif (z <= -4.7e-57)
		tmp = t_0;
	elseif (z <= 27500000.0)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(6.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-6.0 * N[(x * z), $MachinePrecision] + 0.0), $MachinePrecision]}, If[LessEqual[z, -8e+71], t$95$0, If[LessEqual[z, -3250000000000.0], t$95$1, If[LessEqual[z, -4.7e-57], t$95$0, If[LessEqual[z, 27500000.0], x, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.0000000000000003e71 or -3.25e12 < z < -4.6999999999999998e-57

   1. Initial program 99.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

      8. clear-num N/A

         \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{\frac{y \cdot y - x \cdot x}{y + x}}}} \]

      9. flip-- N/A

         \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

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

          \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{y - x}}} \]

      11. --lowering--.f64 99.8

          \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto x + \color{blue}{\frac{6 \cdot z}{\frac{1}{y - x}}} \]

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(6, y \cdot z, 0\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(6, \color{blue}{z \cdot y}, 0\right) \]

      4. *-lowering-*.f64 65.8

         \[\leadsto \mathsf{fma}\left(6, \color{blue}{z \cdot y}, 0\right) \]

   7. Simplified 65.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(6, z \cdot y, 0\right)} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 65.8

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

   9. Applied egg-rr 65.8%

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

   if -8.0000000000000003e71 < z < -3.25e12 or 2.75e7 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-rgt-identity N/A

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

      6. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot x} + -6 \cdot \left(x \cdot z\right)\right) + 0 \]

      7. *-commutative N/A

         \[\leadsto \left(1 \cdot x + -6 \cdot \color{blue}{\left(z \cdot x\right)}\right) + 0 \]

      8. associate-*r* N/A

         \[\leadsto \left(1 \cdot x + \color{blue}{\left(-6 \cdot z\right) \cdot x}\right) + 0 \]

      9. distribute-rgt-in N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot z\right)} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + -6 \cdot z, 0\right)} \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{-6 \cdot z + 1}, 0\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{z \cdot -6} + 1, 0\right) \]

      13. accelerator-lowering-fma.f64 66.8

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(z, -6, 1\right)}, 0\right) \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(z, -6, 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

      5. *-lowering-*.f64 66.8

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

   7. Applied egg-rr 66.8%

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

   8. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

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

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

      5. *-lowering-*.f64 66.3

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

   10. Simplified 66.3%

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

   if -4.6999999999999998e-57 < z < 2.75e7

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 72.8%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+71}:\\ \;\;\;\;6 \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq -3250000000000:\\ \;\;\;\;\mathsf{fma}\left(-6, x \cdot z, 0\right)\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-57}:\\ \;\;\;\;6 \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 27500000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, x \cdot z, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 6 \cdot z, x\right)\\ t_1 := \mathsf{fma}\left(-6, x \cdot z, 0\right)\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2950000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 31500000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (* 6.0 z) x)) (t_1 (fma -6.0 (* x z) 0.0)))
   (if (<= z -7.8e+71)
     t_0
     (if (<= z -2950000000000.0) t_1 (if (<= z 31500000000.0) t_0 t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (6.0 * z), x);
	double t_1 = fma(-6.0, (x * z), 0.0);
	double tmp;
	if (z <= -7.8e+71) {
		tmp = t_0;
	} else if (z <= -2950000000000.0) {
		tmp = t_1;
	} else if (z <= 31500000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(6.0 * z), x)
	t_1 = fma(-6.0, Float64(x * z), 0.0)
	tmp = 0.0
	if (z <= -7.8e+71)
		tmp = t_0;
	elseif (z <= -2950000000000.0)
		tmp = t_1;
	elseif (z <= 31500000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(6.0 * z), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(-6.0 * N[(x * z), $MachinePrecision] + 0.0), $MachinePrecision]}, If[LessEqual[z, -7.8e+71], t$95$0, If[LessEqual[z, -2950000000000.0], t$95$1, If[LessEqual[z, 31500000000.0], t$95$0, t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.8000000000000002e71 or -2.95e12 < z < 3.15e10

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 90.1%

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

   if -7.8000000000000002e71 < z < -2.95e12 or 3.15e10 < z

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-rgt-identity N/A

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

      6. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot x} + -6 \cdot \left(x \cdot z\right)\right) + 0 \]

      7. *-commutative N/A

         \[\leadsto \left(1 \cdot x + -6 \cdot \color{blue}{\left(z \cdot x\right)}\right) + 0 \]

      8. associate-*r* N/A

         \[\leadsto \left(1 \cdot x + \color{blue}{\left(-6 \cdot z\right) \cdot x}\right) + 0 \]

      9. distribute-rgt-in N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot z\right)} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + -6 \cdot z, 0\right)} \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{-6 \cdot z + 1}, 0\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{z \cdot -6} + 1, 0\right) \]

      13. accelerator-lowering-fma.f64 66.8

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(z, -6, 1\right)}, 0\right) \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(z, -6, 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

      5. *-lowering-*.f64 66.8

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

   7. Applied egg-rr 66.8%

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

   8. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

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

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

      5. *-lowering-*.f64 66.3

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

   10. Simplified 66.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-6, x \cdot z, 0\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 6 \cdot z, x\right)\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-6}:\\ \;\;\;\;x + -6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (* 6.0 z) x)))
   (if (<= y -1.2e-72) t_0 (if (<= y 1.16e-6) (+ x (* -6.0 (* x z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (6.0 * z), x);
	double tmp;
	if (y <= -1.2e-72) {
		tmp = t_0;
	} else if (y <= 1.16e-6) {
		tmp = x + (-6.0 * (x * z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(6.0 * z), x)
	tmp = 0.0
	if (y <= -1.2e-72)
		tmp = t_0;
	elseif (y <= 1.16e-6)
		tmp = Float64(x + Float64(-6.0 * Float64(x * z)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(6.0 * z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.2e-72], t$95$0, If[LessEqual[y, 1.16e-6], N[(x + N[(-6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2e-72 or 1.1599999999999999e-6 < y

   1. Initial program 99.7%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 87.6%

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

   if -1.2e-72 < y < 1.1599999999999999e-6

   1. Initial program 99.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

      8. clear-num N/A

         \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{\frac{y \cdot y - x \cdot x}{y + x}}}} \]

      9. flip-- N/A

         \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

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

          \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{y - x}}} \]

      11. --lowering--.f64 99.8

          \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto x + \color{blue}{\frac{6 \cdot z}{\frac{1}{y - x}}} \]

   5. Taylor expanded in y around 0

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. +-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 92.3

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

   7. Simplified 92.3%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(z, -6, 0\right), 0\right)} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

      8. *-lowering-*.f64 92.3

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

   9. Applied egg-rr 92.3%

      \[\leadsto x + \color{blue}{\left(z \cdot x\right) \cdot -6} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-72}:\\ \;\;\;\;\mathsf{fma}\left(y, 6 \cdot z, x\right)\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-6}:\\ \;\;\;\;x + -6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 6 \cdot z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 6 \cdot z, x\right)\\ \mathbf{if}\;y \leq -8.8 \cdot 10^{-73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, -6, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (* 6.0 z) x)))
   (if (<= y -8.8e-73) t_0 (if (<= y 1.9e-6) (fma (* x z) -6.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (6.0 * z), x);
	double tmp;
	if (y <= -8.8e-73) {
		tmp = t_0;
	} else if (y <= 1.9e-6) {
		tmp = fma((x * z), -6.0, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(6.0 * z), x)
	tmp = 0.0
	if (y <= -8.8e-73)
		tmp = t_0;
	elseif (y <= 1.9e-6)
		tmp = fma(Float64(x * z), -6.0, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(6.0 * z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -8.8e-73], t$95$0, If[LessEqual[y, 1.9e-6], N[(N[(x * z), $MachinePrecision] * -6.0 + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.8000000000000001e-73 or 1.9e-6 < y

   1. Initial program 99.7%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 87.6%

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

   if -8.8000000000000001e-73 < y < 1.9e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-rgt-identity N/A

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

      6. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot x} + -6 \cdot \left(x \cdot z\right)\right) + 0 \]

      7. *-commutative N/A

         \[\leadsto \left(1 \cdot x + -6 \cdot \color{blue}{\left(z \cdot x\right)}\right) + 0 \]

      8. associate-*r* N/A

         \[\leadsto \left(1 \cdot x + \color{blue}{\left(-6 \cdot z\right) \cdot x}\right) + 0 \]

      9. distribute-rgt-in N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot z\right)} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + -6 \cdot z, 0\right)} \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{-6 \cdot z + 1}, 0\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{z \cdot -6} + 1, 0\right) \]

      13. accelerator-lowering-fma.f64 92.3

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(z, -6, 1\right)}, 0\right) \]

   5. Simplified 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(z, -6, 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

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

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

      6. *-lowering-*.f64 92.3

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

   7. Applied egg-rr 92.3%

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

Alternative 6: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 6 \cdot z, x\right)\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot -6, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (* 6.0 z) x)))
   (if (<= y -1.4e-72) t_0 (if (<= y 1.95e-6) (fma (* x -6.0) z x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (6.0 * z), x);
	double tmp;
	if (y <= -1.4e-72) {
		tmp = t_0;
	} else if (y <= 1.95e-6) {
		tmp = fma((x * -6.0), z, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(6.0 * z), x)
	tmp = 0.0
	if (y <= -1.4e-72)
		tmp = t_0;
	elseif (y <= 1.95e-6)
		tmp = fma(Float64(x * -6.0), z, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(6.0 * z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.4e-72], t$95$0, If[LessEqual[y, 1.95e-6], N[(N[(x * -6.0), $MachinePrecision] * z + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3999999999999999e-72 or 1.95e-6 < y

   1. Initial program 99.7%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 87.6%

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

   if -1.3999999999999999e-72 < y < 1.95e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-rgt-identity N/A

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

      6. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot x} + -6 \cdot \left(x \cdot z\right)\right) + 0 \]

      7. *-commutative N/A

         \[\leadsto \left(1 \cdot x + -6 \cdot \color{blue}{\left(z \cdot x\right)}\right) + 0 \]

      8. associate-*r* N/A

         \[\leadsto \left(1 \cdot x + \color{blue}{\left(-6 \cdot z\right) \cdot x}\right) + 0 \]

      9. distribute-rgt-in N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot z\right)} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + -6 \cdot z, 0\right)} \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{-6 \cdot z + 1}, 0\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{z \cdot -6} + 1, 0\right) \]

      13. accelerator-lowering-fma.f64 92.3

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(z, -6, 1\right)}, 0\right) \]

   5. Simplified 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(z, -6, 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. *-lowering-*.f64 92.3

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

   7. Applied egg-rr 92.3%

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

Alternative 7: 61.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(z \cdot y\right)\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-53}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 6.0 (* z y))))
   (if (<= z -1.7e-57) t_0 (if (<= z 2.7e-53) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 6.0 * (z * y);
	double tmp;
	if (z <= -1.7e-57) {
		tmp = t_0;
	} else if (z <= 2.7e-53) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 6.0d0 * (z * y)
    if (z <= (-1.7d-57)) then
        tmp = t_0
    else if (z <= 2.7d-53) then
        tmp = x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 6.0 * (z * y);
	double tmp;
	if (z <= -1.7e-57) {
		tmp = t_0;
	} else if (z <= 2.7e-53) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 6.0 * (z * y)
	tmp = 0
	if z <= -1.7e-57:
		tmp = t_0
	elif z <= 2.7e-53:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(6.0 * Float64(z * y))
	tmp = 0.0
	if (z <= -1.7e-57)
		tmp = t_0;
	elseif (z <= 2.7e-53)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 6.0 * (z * y);
	tmp = 0.0;
	if (z <= -1.7e-57)
		tmp = t_0;
	elseif (z <= 2.7e-53)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(6.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.7e-57], t$95$0, If[LessEqual[z, 2.7e-53], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.70000000000000008e-57 or 2.6999999999999999e-53 < z

   1. Initial program 99.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

      8. clear-num N/A

         \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{\frac{y \cdot y - x \cdot x}{y + x}}}} \]

      9. flip-- N/A

         \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

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

          \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{y - x}}} \]

      11. --lowering--.f64 99.7

          \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\frac{6 \cdot z}{\frac{1}{y - x}}} \]

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(6, y \cdot z, 0\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(6, \color{blue}{z \cdot y}, 0\right) \]

      4. *-lowering-*.f64 49.2

         \[\leadsto \mathsf{fma}\left(6, \color{blue}{z \cdot y}, 0\right) \]

   7. Simplified 49.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(6, z \cdot y, 0\right)} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 49.2

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

   9. Applied egg-rr 49.2%

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

   if -1.70000000000000008e-57 < z < 2.6999999999999999e-53

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 76.5%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-57}:\\ \;\;\;\;6 \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-53}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-63}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(6 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.2e-63) (* (* 6.0 z) y) (if (<= z 1.3e-54) x (* z (* 6.0 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.2e-63) {
		tmp = (6.0 * z) * y;
	} else if (z <= 1.3e-54) {
		tmp = x;
	} else {
		tmp = z * (6.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-5.2d-63)) then
        tmp = (6.0d0 * z) * y
    else if (z <= 1.3d-54) then
        tmp = x
    else
        tmp = z * (6.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.2e-63) {
		tmp = (6.0 * z) * y;
	} else if (z <= 1.3e-54) {
		tmp = x;
	} else {
		tmp = z * (6.0 * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.2e-63:
		tmp = (6.0 * z) * y
	elif z <= 1.3e-54:
		tmp = x
	else:
		tmp = z * (6.0 * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.2e-63)
		tmp = Float64(Float64(6.0 * z) * y);
	elseif (z <= 1.3e-54)
		tmp = x;
	else
		tmp = Float64(z * Float64(6.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.2e-63)
		tmp = (6.0 * z) * y;
	elseif (z <= 1.3e-54)
		tmp = x;
	else
		tmp = z * (6.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.2e-63], N[(N[(6.0 * z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 1.3e-54], x, N[(z * N[(6.0 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2000000000000003e-63

   1. Initial program 99.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

      8. clear-num N/A

         \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{\frac{y \cdot y - x \cdot x}{y + x}}}} \]

      9. flip-- N/A

         \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

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

          \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{y - x}}} \]

      11. --lowering--.f64 99.7

          \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\frac{6 \cdot z}{\frac{1}{y - x}}} \]

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(6, y \cdot z, 0\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(6, \color{blue}{z \cdot y}, 0\right) \]

      4. *-lowering-*.f64 55.9

         \[\leadsto \mathsf{fma}\left(6, \color{blue}{z \cdot y}, 0\right) \]

   7. Simplified 55.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(6, z \cdot y, 0\right)} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 55.8

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

   9. Applied egg-rr 55.8%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

       4. *-lowering-*.f64 55.8

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

   11. Applied egg-rr 55.8%

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

   if -5.2000000000000003e-63 < z < 1.30000000000000001e-54

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 76.5%

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

   if 1.30000000000000001e-54 < z

   1. Initial program 99.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

      8. clear-num N/A

         \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{\frac{y \cdot y - x \cdot x}{y + x}}}} \]

      9. flip-- N/A

         \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

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

          \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{y - x}}} \]

      11. --lowering--.f64 99.7

          \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\frac{6 \cdot z}{\frac{1}{y - x}}} \]

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(6, y \cdot z, 0\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(6, \color{blue}{z \cdot y}, 0\right) \]

      4. *-lowering-*.f64 42.3

         \[\leadsto \mathsf{fma}\left(6, \color{blue}{z \cdot y}, 0\right) \]

   7. Simplified 42.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(6, z \cdot y, 0\right)} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 42.3

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

   9. Applied egg-rr 42.3%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-63}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(6 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot z\right) \cdot y\\ \mathbf{if}\;z \leq -4 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* 6.0 z) y)))
   (if (<= z -4e-57) t_0 (if (<= z 6.3e-54) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (6.0 * z) * y;
	double tmp;
	if (z <= -4e-57) {
		tmp = t_0;
	} else if (z <= 6.3e-54) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (6.0d0 * z) * y
    if (z <= (-4d-57)) then
        tmp = t_0
    else if (z <= 6.3d-54) then
        tmp = x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (6.0 * z) * y;
	double tmp;
	if (z <= -4e-57) {
		tmp = t_0;
	} else if (z <= 6.3e-54) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (6.0 * z) * y
	tmp = 0
	if z <= -4e-57:
		tmp = t_0
	elif z <= 6.3e-54:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(6.0 * z) * y)
	tmp = 0.0
	if (z <= -4e-57)
		tmp = t_0;
	elseif (z <= 6.3e-54)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (6.0 * z) * y;
	tmp = 0.0;
	if (z <= -4e-57)
		tmp = t_0;
	elseif (z <= 6.3e-54)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(6.0 * z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -4e-57], t$95$0, If[LessEqual[z, 6.3e-54], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.99999999999999982e-57 or 6.30000000000000002e-54 < z

   1. Initial program 99.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

      8. clear-num N/A

         \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{\frac{y \cdot y - x \cdot x}{y + x}}}} \]

      9. flip-- N/A

         \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

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

          \[\leadsto x + \frac{6 \cdot z}{\color{blue}{\frac{1}{y - x}}} \]

      11. --lowering--.f64 99.7

          \[\leadsto x + \frac{6 \cdot z}{\frac{1}{\color{blue}{y - x}}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\frac{6 \cdot z}{\frac{1}{y - x}}} \]

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(6, y \cdot z, 0\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(6, \color{blue}{z \cdot y}, 0\right) \]

      4. *-lowering-*.f64 49.2

         \[\leadsto \mathsf{fma}\left(6, \color{blue}{z \cdot y}, 0\right) \]

   7. Simplified 49.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(6, z \cdot y, 0\right)} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 49.1

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

   9. Applied egg-rr 49.1%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

       4. *-lowering-*.f64 49.2

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

   11. Applied egg-rr 49.2%

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

   if -3.99999999999999982e-57 < z < 6.30000000000000002e-54

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 76.5%

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

Alternative 10: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. *-lowering-*.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 11: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

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

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

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

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

   7. --lowering--.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 12: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative N/A

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

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

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

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

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

   4. --lowering--.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 13: 36.9% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in z around 0

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

5. Simplified 37.6%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, E"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (* (* 6 z) (- x y))))

  (+ x (* (* (- y x) 6.0) z)))