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

Percentage Accurate: 96.5% → 99.3%

Time: 10.5s

Alternatives: 11

Speedup: 0.8×

• 21.1% 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: 96.5% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-31} \lor \neg \left(z \leq 2 \cdot 10^{-142}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(y, x, -x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1e-31) (not (<= z 2e-142)))
   (fma z (fma y x (- x)) x)
   (fma (* y z) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-31) || !(z <= 2e-142)) {
		tmp = fma(z, fma(y, x, -x), x);
	} else {
		tmp = fma((y * z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1e-31) || !(z <= 2e-142))
		tmp = fma(z, fma(y, x, Float64(-x)), x);
	else
		tmp = fma(Float64(y * z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1e-31], N[Not[LessEqual[z, 2e-142]], $MachinePrecision]], N[(z * N[(y * x + (-x)), $MachinePrecision] + x), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1e-31 or 2.0000000000000001e-142 < z

   1. Initial program 93.8%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 93.8

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.9%

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

   if -1e-31 < z < 2.0000000000000001e-142

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 48.8%

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

   8. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\left(-1 \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot z, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-31} \lor \neg \left(z \leq 2 \cdot 10^{-142}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(y, x, -x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(1 - y\right) \cdot z\\ t_1 := x \cdot \left(\left(-1 + y\right) \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (- 1.0 y) z))) (t_1 (* x (* (+ -1.0 y) z))))
   (if (<= t_0 -2e+41)
     t_1
     (if (<= t_0 2.0)
       (fma (* y z) x x)
       (if (<= t_0 5e+300) t_1 (* (* z x) y))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - ((1.0 - y) * z);
	double t_1 = x * ((-1.0 + y) * z);
	double tmp;
	if (t_0 <= -2e+41) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = fma((y * z), x, x);
	} else if (t_0 <= 5e+300) {
		tmp = t_1;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(Float64(1.0 - y) * z))
	t_1 = Float64(x * Float64(Float64(-1.0 + y) * z))
	tmp = 0.0
	if (t_0 <= -2e+41)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(y * z), x, x);
	elseif (t_0 <= 5e+300)
		tmp = t_1;
	else
		tmp = Float64(Float64(z * x) * y);
	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]}, Block[{t$95$1 = N[(x * N[(N[(-1.0 + y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+41], t$95$1, If[LessEqual[t$95$0, 2.0], N[(N[(y * z), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[t$95$0, 5e+300], t$95$1, N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < -2.00000000000000001e41 or 2 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < 5.00000000000000026e300

   1. Initial program 96.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 96.6

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

   5. Applied rewrites 96.6%

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

   if -2.00000000000000001e41 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < 2

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. Step-by-step derivation

   7. Applied rewrites 47.7%

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

   8. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\left(-1 \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot z, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 98.4%

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

   if 5.00000000000000026e300 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z))

   1. Initial program 61.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(1 - y\right) \cdot z \leq -2 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \left(\left(-1 + y\right) \cdot z\right)\\ \mathbf{elif}\;1 - \left(1 - y\right) \cdot z \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{elif}\;1 - \left(1 - y\right) \cdot z \leq 5 \cdot 10^{+300}:\\ \;\;\;\;x \cdot \left(\left(-1 + y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+227}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{+46} \lor \neg \left(y \leq 8 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.5e+227)
   (* (* z x) y)
   (if (or (<= y -1.2e+46) (not (<= y 8e-10)))
     (fma (* y z) x x)
     (fma (- z) x x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.5e+227) {
		tmp = (z * x) * y;
	} else if ((y <= -1.2e+46) || !(y <= 8e-10)) {
		tmp = fma((y * z), x, x);
	} else {
		tmp = fma(-z, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.5e+227)
		tmp = Float64(Float64(z * x) * y);
	elseif ((y <= -1.2e+46) || !(y <= 8e-10))
		tmp = fma(Float64(y * z), x, x);
	else
		tmp = fma(Float64(-z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.5e+227], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[Or[LessEqual[y, -1.2e+46], N[Not[LessEqual[y, 8e-10]], $MachinePrecision]], N[(N[(y * z), $MachinePrecision] * x + x), $MachinePrecision], N[((-z) * x + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.4999999999999995e227

   1. Initial program 66.1%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

   if -8.4999999999999995e227 < y < -1.20000000000000004e46 or 8.00000000000000029e-10 < y

   1. Initial program 94.5%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 94.6

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 62.7%

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

   8. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\left(-1 \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot z, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 94.5%

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

   if -1.20000000000000004e46 < y < 8.00000000000000029e-10

   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. Step-by-step derivation

   7. Applied rewrites 41.5%

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

   8. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\left(-1 \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot z, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.7%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 99.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+227}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{+46} \lor \neg \left(y \leq 8 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 25.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq -5 \cdot 10^{-166}:\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x (- 1.0 (* (- 1.0 y) z))) -5e-166) (* z x) (* x 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * (1.0 - ((1.0 - y) * z))) <= -5e-166) {
		tmp = z * x;
	} 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 ((x * (1.0d0 - ((1.0d0 - y) * z))) <= (-5d-166)) then
        tmp = z * x
    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 ((x * (1.0 - ((1.0 - y) * z))) <= -5e-166) {
		tmp = z * x;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * (1.0 - ((1.0 - y) * z))) <= -5e-166:
		tmp = z * x
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z))) <= -5e-166)
		tmp = Float64(z * x);
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * (1.0 - ((1.0 - y) * z))) <= -5e-166)
		tmp = z * x;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-166], N[(z * x), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z))) < -5e-166

   1. Initial program 98.1%

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

   4. Applied rewrites 59.7%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 41.6

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

   7. Applied rewrites 41.6%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 9.2%

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

   if -5e-166 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)))

   1. Initial program 94.2%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 49.6

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

   5. Applied rewrites 49.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 47.0%

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

Alternative 5: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+70} \lor \neg \left(y \leq 370\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.25e+70) (not (<= y 370.0))) (* (* y x) z) (fma (- z) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.25e+70) || !(y <= 370.0)) {
		tmp = (y * x) * z;
	} else {
		tmp = fma(-z, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.25e+70) || !(y <= 370.0))
		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, -1.25e+70], N[Not[LessEqual[y, 370.0]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], N[((-z) * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2500000000000001e70 or 370 < y

   1. Initial program 90.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 75.9

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

   8. Applied rewrites 75.9%

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

   if -1.2500000000000001e70 < y < 370

   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. Step-by-step derivation

   7. Applied rewrites 41.0%

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

   8. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\left(-1 \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot z, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 60.8%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 98.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+70} \lor \neg \left(y \leq 370\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(y, x, -x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-1 + y\right) \cdot z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4e+125) (fma z (fma y x (- x)) x) (fma (* (+ -1.0 y) z) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4e+125) {
		tmp = fma(z, fma(y, x, -x), x);
	} else {
		tmp = fma(((-1.0 + y) * z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4e+125)
		tmp = fma(z, fma(y, x, Float64(-x)), x);
	else
		tmp = fma(Float64(Float64(-1.0 + y) * z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4e+125], N[(z * N[(y * x + (-x)), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(-1.0 + y), $MachinePrecision] * z), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999997e125

   1. Initial program 95.0%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 96.8%

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

   if 3.9999999999999997e125 < x

   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%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(y, x, -x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-1 + y\right) \cdot z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -660 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -660.0) (not (<= z 1.0))) (* x (- z)) (* x 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -660.0) || !(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 <= (-660.0d0)) .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 <= -660.0) || !(z <= 1.0)) {
		tmp = x * -z;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -660.0) 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 <= -660.0) || !(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 <= -660.0) || ~((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, -660.0], 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 < -660 or 1 < z

   1. Initial program 91.6%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

   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 50.7%

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

   if -660 < 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 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 22.3

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

   5. Applied rewrites 22.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 79.3%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -660 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8e-10) (fma (- z) x x) (fma z x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8e-10) {
		tmp = fma(-z, x, x);
	} else {
		tmp = fma(z, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8e-10)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = fma(z, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8e-10], N[((-z) * x + x), $MachinePrecision], N[(z * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.00000000000000029e-10

   1. Initial program 97.3%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 30.6%

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

   8. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\left(-1 \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot z, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 67.6%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 80.8%

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

   if 8.00000000000000029e-10 < y

   1. Initial program 92.1%

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 46.4

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

   7. Applied rewrites 46.4%

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

4. Final simplification 71.0%

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

Alternative 9: 69.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-10}:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8e-10) (- x (* z x)) (fma z x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8e-10) {
		tmp = x - (z * x);
	} else {
		tmp = fma(z, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8e-10)
		tmp = Float64(x - Float64(z * x));
	else
		tmp = fma(z, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8e-10], N[(x - N[(z * x), $MachinePrecision]), $MachinePrecision], N[(z * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.00000000000000029e-10

   1. Initial program 97.3%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.8%

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

   if 8.00000000000000029e-10 < y

   1. Initial program 92.1%

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 46.4

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

   7. Applied rewrites 46.4%

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

4. Final simplification 71.0%

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

Alternative 10: 41.3% accurate, 2.4× 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(z, x, x)
end

Wolfram

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

Derivation

1. Initial program 95.8%

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

4. Applied rewrites 63.0%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f64 46.1

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

7. Applied rewrites 46.1%

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

Alternative 11: 6.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.8%

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

4. Applied rewrites 63.0%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f64 46.1

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

7. Applied rewrites 46.1%

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

8. Taylor expanded in z around inf

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

10. Applied rewrites 7.5%

    \[\leadsto z \cdot \color{blue}{x} \]
11. 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 2024337 
(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))))