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

Percentage Accurate: 96.2% → 98.2%

Time: 7.4s

Alternatives: 7

Speedup: 0.5×

• 21.2% 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: 96.2% 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: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < -inf.0

   1. Initial program 73.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z))

   1. Initial program 98.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 2: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(1 - y\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq -10000000 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;x \cdot \left(\left(-1 + y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (- 1.0 y) z))))
   (if (<= t_0 (- INFINITY))
     (* (* y x) z)
     (if (or (<= t_0 -10000000.0) (not (<= t_0 2.0)))
       (* x (* (+ -1.0 y) z))
       (fma (- z) x x)))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - ((1.0 - y) * z);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y * x) * z;
	} else if ((t_0 <= -10000000.0) || !(t_0 <= 2.0)) {
		tmp = x * ((-1.0 + y) * z);
	} else {
		tmp = fma(-z, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(Float64(1.0 - y) * z))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(y * x) * z);
	elseif ((t_0 <= -10000000.0) || !(t_0 <= 2.0))
		tmp = Float64(x * Float64(Float64(-1.0 + y) * z));
	else
		tmp = fma(Float64(-z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[Or[LessEqual[t$95$0, -10000000.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(x * N[(N[(-1.0 + y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[((-z) * x + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < -inf.0

   1. Initial program 73.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < -1e7 or 2 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z))

   1. Initial program 97.1%

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

   3. Taylor expanded in z around inf

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

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

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-rgt-in N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

      19. fp-cancel-sign-sub-inv N/A

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

      20. distribute-neg-in N/A

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

      21. metadata-eval N/A

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

      22. mul-1-neg N/A

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

      23. remove-double-neg N/A

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

      24. lower-+.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if -1e7 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.2%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(1 - y\right) \cdot z \leq -\infty:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;1 - \left(1 - y\right) \cdot z \leq -10000000 \lor \neg \left(1 - \left(1 - y\right) \cdot z \leq 2\right):\\ \;\;\;\;x \cdot \left(\left(-1 + y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+116} \lor \neg \left(y \leq 1.15 \cdot 10^{+101}\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.1e+116) (not (<= y 1.15e+101)))
   (* (* y x) z)
   (fma (- z) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.1e+116) || !(y <= 1.15e+101)) {
		tmp = (y * x) * z;
	} else {
		tmp = fma(-z, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.1e+116) || !(y <= 1.15e+101))
		tmp = Float64(Float64(y * x) * z);
	else
		tmp = fma(Float64(-z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.1e+116], N[Not[LessEqual[y, 1.15e+101]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], N[((-z) * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.09999999999999996e116 or 1.1500000000000001e101 < y

   1. Initial program 90.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 80.0

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

   5. Applied rewrites 80.0%

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

   7. Applied rewrites 81.0%

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

   if -3.09999999999999996e116 < y < 1.1500000000000001e101

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 94.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+116} \lor \neg \left(y \leq 1.15 \cdot 10^{+101}\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+116}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.1e+116)
   (* (* y x) z)
   (if (<= y 1.15e+101) (fma (- z) x x) (* (* z x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e+116) {
		tmp = (y * x) * z;
	} else if (y <= 1.15e+101) {
		tmp = fma(-z, x, x);
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.1e+116)
		tmp = Float64(Float64(y * x) * z);
	elseif (y <= 1.15e+101)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = Float64(Float64(z * x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.1e+116], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[y, 1.15e+101], N[((-z) * x + x), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.09999999999999996e116

   1. Initial program 95.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

   7. Applied rewrites 79.2%

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

   if -3.09999999999999996e116 < y < 1.1500000000000001e101

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 94.7%

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

   if 1.1500000000000001e101 < y

   1. Initial program 87.3%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

Alternative 5: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0))) (* x (- z)) (* x 1.0)))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x * -z
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = x * -z;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = x * -z
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(x * Float64(-z));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = x * -z;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x * (-z)), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 93.3%

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

   3. Taylor expanded in z around inf

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

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

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-rgt-in N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

      19. fp-cancel-sign-sub-inv N/A

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

      20. distribute-neg-in N/A

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

      21. metadata-eval N/A

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

      22. mul-1-neg N/A

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

      23. remove-double-neg N/A

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

      24. lower-+.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.4%

      \[\leadsto x \cdot \left(-z\right) \]

   if -1 < z < 1

   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

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

      1. lower--.f64 78.4

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

   5. Applied rewrites 78.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 76.5%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.9% 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 96.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. lower-fma.f64 N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. lower-*.f64 N/A

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

   11. mul-1-neg N/A

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

   12. *-lft-identity N/A

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

   13. metadata-eval N/A

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

   14. fp-cancel-sign-sub-inv N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. mul-1-neg N/A

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

   18. remove-double-neg N/A

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

   19. lower-+.f64 96.6

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

5. Applied rewrites 96.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 70.4%

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

Alternative 7: 37.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
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 70.4

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

5. Applied rewrites 70.4%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 40.0%

   \[\leadsto x \cdot \color{blue}{1} \]
9. 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 2024340 
(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))))