Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3

Percentage Accurate: 97.9% → 100.0%

Time: 7.6s

Alternatives: 6

Speedup: 1.7×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-in N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. mul-1-neg N/A

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

   11. remove-double-neg N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 61.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-15}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 0.0052:\\ \;\;\;\;\mathsf{fma}\left(z, x, z\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+207}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= x -1.3e+159)
     t_0
     (if (<= x -3.2e-15)
       (* x y)
       (if (<= x 0.0052) (fma z x z) (if (<= x 3.8e+207) (* x y) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (x <= -1.3e+159) {
		tmp = t_0;
	} else if (x <= -3.2e-15) {
		tmp = x * y;
	} else if (x <= 0.0052) {
		tmp = fma(z, x, z);
	} else if (x <= 3.8e+207) {
		tmp = x * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (x <= -1.3e+159)
		tmp = t_0;
	elseif (x <= -3.2e-15)
		tmp = Float64(x * y);
	elseif (x <= 0.0052)
		tmp = fma(z, x, z);
	elseif (x <= 3.8e+207)
		tmp = Float64(x * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[x, -1.3e+159], t$95$0, If[LessEqual[x, -3.2e-15], N[(x * y), $MachinePrecision], If[LessEqual[x, 0.0052], N[(z * x + z), $MachinePrecision], If[LessEqual[x, 3.8e+207], N[(x * y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3e159 or 3.79999999999999986e207 < x

   1. Initial program 90.4%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 66.5

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

   8. Applied rewrites 66.5%

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

   if -1.3e159 < x < -3.1999999999999999e-15 or 0.0051999999999999998 < x < 3.79999999999999986e207

   1. Initial program 97.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   if -3.1999999999999999e-15 < x < 0.0051999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 78.1

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

   7. Applied rewrites 78.1%

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

      1. neg-sub0 N/A

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

      2. flip3-- N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{0}^{3} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}}, x, z\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left({z}^{3}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      5. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{z}^{\left(\frac{3}{2}\right)} \cdot {z}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(z \cdot z\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      7. sqr-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      9. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      10. pow-prod-down N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      11. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{3}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}^{3}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      13. cube-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({z}^{3}\right)\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      14. sub0-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(0 - {z}^{3}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\color{blue}{{0}^{3}} - {z}^{3}\right)\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\frac{{0}^{3} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)}, x, z\right) \]

      17. flip3-- N/A

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

      18. neg-sub0 N/A

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

      19. remove-double-neg 77.0

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

   9. Applied rewrites 77.0%

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

Alternative 3: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;z - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))))
   (if (<= x -1.2e-8) t_0 (if (<= x 0.23) (- z (* x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -1.2e-8) {
		tmp = t_0;
	} else if (x <= 0.23) {
		tmp = z - (x * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y - z)
    if (x <= (-1.2d-8)) then
        tmp = t_0
    else if (x <= 0.23d0) then
        tmp = z - (x * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -1.2e-8) {
		tmp = t_0;
	} else if (x <= 0.23) {
		tmp = z - (x * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y - z)
	tmp = 0
	if x <= -1.2e-8:
		tmp = t_0
	elif x <= 0.23:
		tmp = z - (x * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y - z))
	tmp = 0.0
	if (x <= -1.2e-8)
		tmp = t_0;
	elseif (x <= 0.23)
		tmp = Float64(z - Float64(x * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y - z);
	tmp = 0.0;
	if (x <= -1.2e-8)
		tmp = t_0;
	elseif (x <= 0.23)
		tmp = z - (x * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e-8], t$95$0, If[LessEqual[x, 0.23], N[(z - N[(x * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.19999999999999999e-8 or 0.23000000000000001 < x

   1. Initial program 94.7%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -1.19999999999999999e-8 < x < 0.23000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;z - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.0052:\\ \;\;\;\;\mathsf{fma}\left(z, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))))
   (if (<= x -3.1e-15) t_0 (if (<= x 0.0052) (fma z x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -3.1e-15) {
		tmp = t_0;
	} else if (x <= 0.0052) {
		tmp = fma(z, x, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y - z))
	tmp = 0.0
	if (x <= -3.1e-15)
		tmp = t_0;
	elseif (x <= 0.0052)
		tmp = fma(z, x, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.1e-15], t$95$0, If[LessEqual[x, 0.0052], N[(z * x + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.0999999999999999e-15 or 0.0051999999999999998 < x

   1. Initial program 94.7%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -3.0999999999999999e-15 < x < 0.0051999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 78.1

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

   7. Applied rewrites 78.1%

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

      1. neg-sub0 N/A

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

      2. flip3-- N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{0}^{3} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}}, x, z\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left({z}^{3}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      5. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{z}^{\left(\frac{3}{2}\right)} \cdot {z}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(z \cdot z\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      7. sqr-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      9. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      10. pow-prod-down N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      11. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{3}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}^{3}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      13. cube-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({z}^{3}\right)\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      14. sub0-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(0 - {z}^{3}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\color{blue}{{0}^{3}} - {z}^{3}\right)\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\frac{{0}^{3} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)}, x, z\right) \]

      17. flip3-- N/A

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

      18. neg-sub0 N/A

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

      19. remove-double-neg 77.0

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

   9. Applied rewrites 77.0%

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

Alternative 5: 61.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-15}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 0.0052:\\ \;\;\;\;\mathsf{fma}\left(z, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.2e-15) (* x y) (if (<= x 0.0052) (fma z x z) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.2e-15) {
		tmp = x * y;
	} else if (x <= 0.0052) {
		tmp = fma(z, x, z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.2e-15)
		tmp = Float64(x * y);
	elseif (x <= 0.0052)
		tmp = fma(z, x, z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.2e-15], N[(x * y), $MachinePrecision], If[LessEqual[x, 0.0052], N[(z * x + z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1999999999999999e-15 or 0.0051999999999999998 < x

   1. Initial program 94.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

   if -3.1999999999999999e-15 < x < 0.0051999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 78.1

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

   7. Applied rewrites 78.1%

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

      1. neg-sub0 N/A

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

      2. flip3-- N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{0}^{3} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}}, x, z\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left({z}^{3}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      5. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{z}^{\left(\frac{3}{2}\right)} \cdot {z}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(z \cdot z\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      7. sqr-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      9. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      10. pow-prod-down N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      11. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{3}}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}^{3}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      13. cube-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({z}^{3}\right)\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      14. sub0-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(0 - {z}^{3}\right)}\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\color{blue}{{0}^{3}} - {z}^{3}\right)\right)}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}, x, z\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\frac{{0}^{3} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)}, x, z\right) \]

      17. flip3-- N/A

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

      18. neg-sub0 N/A

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

      19. remove-double-neg 77.0

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

   9. Applied rewrites 77.0%

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

Alternative 6: 41.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 40.2

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

5. Applied rewrites 40.2%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))