Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J

Percentage Accurate: 95.8% → 99.3%

Time: 11.8s

Alternatives: 13

Speedup: 0.8×

• 20.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * 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 * (1.0d0 - ((1.0d0 - y) * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * 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 * (1.0d0 - ((1.0d0 - y) * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot \left(y + -1\right), z, x\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-158}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* x (+ y -1.0)) z x)))
   (if (<= z -2e-19) t_0 (if (<= z 1.2e-158) (fma (* y z) x x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x * (y + -1.0)), z, x);
	double tmp;
	if (z <= -2e-19) {
		tmp = t_0;
	} else if (z <= 1.2e-158) {
		tmp = fma((y * z), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x * Float64(y + -1.0)), z, x)
	tmp = 0.0
	if (z <= -2e-19)
		tmp = t_0;
	elseif (z <= 1.2e-158)
		tmp = fma(Float64(y * z), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -2e-19], t$95$0, If[LessEqual[z, 1.2e-158], N[(N[(y * z), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e-19 or 1.20000000000000004e-158 < z

   1. Initial program 95.7%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 100.0%

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

   if -2e-19 < z < 1.20000000000000004e-158

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 95.1%

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

      1. lift-+.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.9

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

   9. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y + -1\right), z, x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-158}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y + -1\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{if}\;1 - y \leq -500000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* y z) x x)))
   (if (<= (- 1.0 y) -500000000.0)
     t_0
     (if (<= (- 1.0 y) 1.0) (fma (- z) x x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((y * z), x, x);
	double tmp;
	if ((1.0 - y) <= -500000000.0) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1.0) {
		tmp = fma(-z, x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(y * z), x, x)
	tmp = 0.0
	if (Float64(1.0 - y) <= -500000000.0)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 1.0)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * z), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[N[(1.0 - y), $MachinePrecision], -500000000.0], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 1.0], N[((-z) * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -5e8 or 1 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 94.3%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 95.5%

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

      1. lift-+.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 94.3

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

   6. Applied egg-rr 94.3%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 93.7

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

   9. Simplified 93.7%

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

   if -5e8 < (-.f64 #s(literal 1 binary64) y) < 1

   1. Initial program 100.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 98.9

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

   5. Simplified 98.9%

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

      1. cancel-sign-sub-inv N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 98.9

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -500000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{elif}\;1 - y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot y, z, x\right)\\ \mathbf{if}\;1 - y \leq -500000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* x y) z x)))
   (if (<= (- 1.0 y) -500000000.0)
     t_0
     (if (<= (- 1.0 y) 1.0) (fma (- z) x x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x * y), z, x);
	double tmp;
	if ((1.0 - y) <= -500000000.0) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1.0) {
		tmp = fma(-z, x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x * y), z, x)
	tmp = 0.0
	if (Float64(1.0 - y) <= -500000000.0)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 1.0)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[N[(1.0 - y), $MachinePrecision], -500000000.0], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 1.0], N[((-z) * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -5e8 or 1 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 94.3%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 88.2%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 87.5

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

   7. Simplified 87.5%

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

   if -5e8 < (-.f64 #s(literal 1 binary64) y) < 1

   1. Initial program 100.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 98.9

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

   5. Simplified 98.9%

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

      1. cancel-sign-sub-inv N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 98.9

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

   7. Applied egg-rr 98.9%

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

Alternative 4: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -2 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;1 - y \leq 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- 1.0 y) -2e+45)
   (* y (* x z))
   (if (<= (- 1.0 y) 1e+91) (fma (- z) x x) (* x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - y) <= -2e+45) {
		tmp = y * (x * z);
	} else if ((1.0 - y) <= 1e+91) {
		tmp = fma(-z, x, x);
	} else {
		tmp = x * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 - y) <= -2e+45)
		tmp = Float64(y * Float64(x * z));
	elseif (Float64(1.0 - y) <= 1e+91)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = Float64(x * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 - y), $MachinePrecision], -2e+45], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 - y), $MachinePrecision], 1e+91], N[((-z) * x + x), $MachinePrecision], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1.9999999999999999e45

   1. Initial program 91.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 72.5

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

   5. Simplified 72.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 76.1

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

   7. Applied egg-rr 76.1%

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

   if -1.9999999999999999e45 < (-.f64 #s(literal 1 binary64) y) < 1.00000000000000008e91

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.4

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

   5. Simplified 95.4%

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

      1. cancel-sign-sub-inv N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 95.4

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

   7. Applied egg-rr 95.4%

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

   if 1.00000000000000008e91 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 96.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.4

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

   5. Simplified 86.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -2 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;1 - y \leq 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -2 \cdot 10^{+45}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;1 - y \leq 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- 1.0 y) -2e+45)
   (* z (* x y))
   (if (<= (- 1.0 y) 1e+91) (fma (- z) x x) (* x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - y) <= -2e+45) {
		tmp = z * (x * y);
	} else if ((1.0 - y) <= 1e+91) {
		tmp = fma(-z, x, x);
	} else {
		tmp = x * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 - y) <= -2e+45)
		tmp = Float64(z * Float64(x * y));
	elseif (Float64(1.0 - y) <= 1e+91)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = Float64(x * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 - y), $MachinePrecision], -2e+45], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 - y), $MachinePrecision], 1e+91], N[((-z) * x + x), $MachinePrecision], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1.9999999999999999e45

   1. Initial program 91.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 72.5

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

   5. Simplified 72.5%

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

   if -1.9999999999999999e45 < (-.f64 #s(literal 1 binary64) y) < 1.00000000000000008e91

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.4

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

   5. Simplified 95.4%

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

      1. cancel-sign-sub-inv N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 95.4

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

   7. Applied egg-rr 95.4%

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

   if 1.00000000000000008e91 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 96.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.4

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

   5. Simplified 86.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -2 \cdot 10^{+45}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;1 - y \leq 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(x \cdot y - x\right)\\ \mathbf{if}\;z \leq -65:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* x y) x))))
   (if (<= z -65.0) t_0 (if (<= z 2.5e-7) (fma (* y z) x x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((x * y) - x);
	double tmp;
	if (z <= -65.0) {
		tmp = t_0;
	} else if (z <= 2.5e-7) {
		tmp = fma((y * z), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(x * y) - x))
	tmp = 0.0
	if (z <= -65.0)
		tmp = t_0;
	elseif (z <= 2.5e-7)
		tmp = fma(Float64(y * z), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -65.0], t$95$0, If[LessEqual[z, 2.5e-7], N[(N[(y * z), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -65 or 2.49999999999999989e-7 < z

   1. Initial program 94.5%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-out-- N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

      17. lower-*.f64 N/A

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

      18. distribute-rgt-out-- N/A

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

      19. *-lft-identity N/A

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

      20. unsub-neg N/A

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

      21. +-commutative N/A

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

      22. distribute-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. unsub-neg N/A

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

      25. lower--.f64 N/A

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

      26. lower-*.f64 99.0

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

   5. Simplified 99.0%

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

   if -65 < z < 2.49999999999999989e-7

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 96.0%

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

      1. lift-+.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.5

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

   9. Simplified 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -65:\\ \;\;\;\;z \cdot \left(x \cdot y - x\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;y \leq -1.36 \cdot 10^{+91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* y z))))
   (if (<= y -1.36e+91) t_0 (if (<= y 1.7e+31) (fma (- z) x x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y * z);
	double tmp;
	if (y <= -1.36e+91) {
		tmp = t_0;
	} else if (y <= 1.7e+31) {
		tmp = fma(-z, x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y * z))
	tmp = 0.0
	if (y <= -1.36e+91)
		tmp = t_0;
	elseif (y <= 1.7e+31)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.36e+91], t$95$0, If[LessEqual[y, 1.7e+31], N[((-z) * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.36000000000000007e91 or 1.6999999999999999e31 < y

   1. Initial program 93.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.9

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

   5. Simplified 78.9%

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

   if -1.36000000000000007e91 < y < 1.6999999999999999e31

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.4

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

   5. Simplified 95.4%

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

      1. cancel-sign-sub-inv N/A

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

      2. distribute-rgt1-in N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 95.4

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

   7. Applied egg-rr 95.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.36 \cdot 10^{+91}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y + -1\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y + -1\right) \cdot z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4e-47) (fma (* x (+ y -1.0)) z x) (fma (* (+ y -1.0) z) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4e-47) {
		tmp = fma((x * (y + -1.0)), z, x);
	} else {
		tmp = fma(((y + -1.0) * z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4e-47)
		tmp = fma(Float64(x * Float64(y + -1.0)), z, x);
	else
		tmp = fma(Float64(Float64(y + -1.0) * z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4e-47], N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(N[(y + -1.0), $MachinePrecision] * z), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999999e-47

   1. Initial program 96.3%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 95.4%

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

   if 3.9999999999999999e-47 < x

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \left(y + -1\right), x, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y + -1\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y + -1\right) \cdot z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -65:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- x)))) (if (<= z -65.0) t_0 (if (<= z 1.25e-5) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (z <= -65.0) {
		tmp = t_0;
	} else if (z <= 1.25e-5) {
		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 = z * -x
    if (z <= (-65.0d0)) then
        tmp = t_0
    else if (z <= 1.25d-5) 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 = z * -x;
	double tmp;
	if (z <= -65.0) {
		tmp = t_0;
	} else if (z <= 1.25e-5) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -x
	tmp = 0
	if z <= -65.0:
		tmp = t_0
	elif z <= 1.25e-5:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-x))
	tmp = 0.0
	if (z <= -65.0)
		tmp = t_0;
	elseif (z <= 1.25e-5)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * -x;
	tmp = 0.0;
	if (z <= -65.0)
		tmp = t_0;
	elseif (z <= 1.25e-5)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * (-x)), $MachinePrecision]}, If[LessEqual[z, -65.0], t$95$0, If[LessEqual[z, 1.25e-5], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -65 or 1.25000000000000006e-5 < z

   1. Initial program 94.5%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 61.8

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

   5. Simplified 61.8%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 60.9

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

   8. Simplified 60.9%

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

   if -65 < z < 1.25000000000000006e-5

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

   5. Simplified 73.8%

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

      1. *-rgt-identity 73.8

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

   7. Applied egg-rr 73.8%

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

Alternative 10: 66.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.3%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. distribute-lft-out-- N/A

      \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

   2. *-rgt-identity N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 68.2

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

5. Simplified 68.2%

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

   1. cancel-sign-sub-inv N/A

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

   2. distribute-rgt1-in N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 68.3

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

7. Applied egg-rr 68.3%

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

Alternative 11: 66.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. distribute-lft-out-- N/A

      \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

   2. *-rgt-identity N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 68.2

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

5. Simplified 68.2%

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

6. Final simplification 68.2%

   \[\leadsto x - x \cdot z \]
7. Add Preprocessing

Alternative 12: 66.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - z\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x * (1.0 - 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 * (1.0d0 - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. lower--.f64 68.2

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

5. Simplified 68.2%

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

Alternative 13: 39.1% 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 97.3%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

5. Simplified 39.8%

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

   1. *-rgt-identity 39.8

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

7. Applied egg-rr 39.8%

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

Developer Target 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ t_1 := x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{if}\;t\_0 < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* (- 1.0 y) z))))
        (t_1 (+ x (* (- 1.0 y) (* (- z) x)))))
   (if (< t_0 -1.618195973607049e+50)
     t_1
     (if (< t_0 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = x * (1.0 - ((1.0 - y) * z));
	double t_1 = x + ((1.0 - y) * (-z * x));
	double tmp;
	if (t_0 < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (t_0 < 3.892237649663903e+134) {
		tmp = ((x * y) * z) - ((x * z) - x);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * (1.0d0 - ((1.0d0 - y) * z))
    t_1 = x + ((1.0d0 - y) * (-z * x))
    if (t_0 < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (t_0 < 3.892237649663903d+134) then
        tmp = ((x * y) * z) - ((x * z) - x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (1.0 - ((1.0 - y) * z));
	double t_1 = x + ((1.0 - y) * (-z * x));
	double tmp;
	if (t_0 < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (t_0 < 3.892237649663903e+134) {
		tmp = ((x * y) * z) - ((x * z) - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (1.0 - ((1.0 - y) * z))
	t_1 = x + ((1.0 - y) * (-z * x))
	tmp = 0
	if t_0 < -1.618195973607049e+50:
		tmp = t_1
	elif t_0 < 3.892237649663903e+134:
		tmp = ((x * y) * z) - ((x * z) - x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
	t_1 = Float64(x + Float64(Float64(1.0 - y) * Float64(Float64(-z) * x)))
	tmp = 0.0
	if (t_0 < -1.618195973607049e+50)
		tmp = t_1;
	elseif (t_0 < 3.892237649663903e+134)
		tmp = Float64(Float64(Float64(x * y) * z) - Float64(Float64(x * z) - x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (1.0 - ((1.0 - y) * z));
	t_1 = x + ((1.0 - y) * (-z * x));
	tmp = 0.0;
	if (t_0 < -1.618195973607049e+50)
		tmp = t_1;
	elseif (t_0 < 3.892237649663903e+134)
		tmp = ((x * y) * z) - ((x * z) - x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(N[(1.0 - y), $MachinePrecision] * N[((-z) * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$0, -1.618195973607049e+50], t$95$1, If[Less[t$95$0, 3.892237649663903e+134], N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] - N[(N[(x * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024208 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* x (- 1 (* (- 1 y) z))) -161819597360704900000000000000000000000000000000000) (+ x (* (- 1 y) (* (- z) x))) (if (< (* x (- 1 (* (- 1 y) z))) 389223764966390300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1 y) (* (- z) x))))))

  (* x (- 1.0 (* (- 1.0 y) z))))