Diagrams.TwoD.Segment:bezierClip from diagrams-lib-1.3.0.3

Percentage Accurate: 98.1% → 100.0%

Time: 6.6s

Alternatives: 6

Speedup: 1.7×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in x around 0

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

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

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

   2. unsub-neg N/A

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

   3. *-lft-identity N/A

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. *-commutative N/A

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

   8. mul-1-neg N/A

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

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

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

   10. mul-1-neg N/A

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

   11. distribute-lft-in N/A

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

   12. +-commutative N/A

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

   13. remove-double-neg N/A

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

   14. mul-1-neg N/A

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

   15. mul-1-neg N/A

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

   16. distribute-neg-in N/A

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

   17. lower-fma.f64 N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 60.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-y\right)\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-98}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(z, y, z\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+96}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- y))))
   (if (<= y -3.3e+156)
     t_0
     (if (<= y -8e-98)
       (* y x)
       (if (<= y 4.3e-7) (fma z y z) (if (<= y 8.5e+96) (* y x) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -y;
	double tmp;
	if (y <= -3.3e+156) {
		tmp = t_0;
	} else if (y <= -8e-98) {
		tmp = y * x;
	} else if (y <= 4.3e-7) {
		tmp = fma(z, y, z);
	} else if (y <= 8.5e+96) {
		tmp = y * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-y))
	tmp = 0.0
	if (y <= -3.3e+156)
		tmp = t_0;
	elseif (y <= -8e-98)
		tmp = Float64(y * x);
	elseif (y <= 4.3e-7)
		tmp = fma(z, y, z);
	elseif (y <= 8.5e+96)
		tmp = Float64(y * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * (-y)), $MachinePrecision]}, If[LessEqual[y, -3.3e+156], t$95$0, If[LessEqual[y, -8e-98], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.3e-7], N[(z * y + z), $MachinePrecision], If[LessEqual[y, 8.5e+96], N[(y * x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2999999999999999e156 or 8.50000000000000025e96 < y

   1. Initial program 92.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 67.3

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

   8. Applied rewrites 67.3%

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

   if -3.2999999999999999e156 < y < -7.99999999999999951e-98 or 4.3000000000000001e-7 < y < 8.50000000000000025e96

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 71.0

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

   5. Applied rewrites 71.0%

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

   if -7.99999999999999951e-98 < y < 4.3000000000000001e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. neg-sub0 N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. sub0-neg N/A

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

      5. cube-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. sqr-pow N/A

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

      8. unpow-prod-down N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqr-neg N/A

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

      12. pow-prod-down N/A

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

      13. sqr-pow N/A

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

      14. metadata-eval N/A

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

      15. +-lft-identity N/A

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

      16. distribute-rgt-out N/A

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

      17. +-commutative N/A

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

      18. +-lft-identity N/A

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

      19. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{z}^{3}}{\color{blue}{{z}^{2}}}, y, z\right) \]

      20. pow-div N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{z}^{\left(3 - 2\right)}}, y, z\right) \]

      21. metadata-eval N/A

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

      22. unpow1 73.8

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

   9. Applied rewrites 73.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+156}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-98}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(z, y, z\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+96}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, z\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-65}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.9e-133)
   (fma (- z) y z)
   (if (<= z 6.5e-65) (* y x) (- z (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.9e-133) {
		tmp = fma(-z, y, z);
	} else if (z <= 6.5e-65) {
		tmp = y * x;
	} else {
		tmp = z - (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.9e-133)
		tmp = fma(Float64(-z), y, z);
	elseif (z <= 6.5e-65)
		tmp = Float64(y * x);
	else
		tmp = Float64(z - Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.9e-133], N[((-z) * y + z), $MachinePrecision], If[LessEqual[z, 6.5e-65], N[(y * x), $MachinePrecision], N[(z - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8999999999999998e-133

   1. Initial program 96.2%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 82.2

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

   7. Applied rewrites 82.2%

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

   if -2.8999999999999998e-133 < z < 6.5e-65

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if 6.5e-65 < z

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, z\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-65}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z - y \cdot z\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{-133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-65}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- z (* y z))))
   (if (<= z -2.9e-133) t_0 (if (<= z 6.5e-65) (* y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z - (y * z);
	double tmp;
	if (z <= -2.9e-133) {
		tmp = t_0;
	} else if (z <= 6.5e-65) {
		tmp = y * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z - (y * z)
    if (z <= (-2.9d-133)) then
        tmp = t_0
    else if (z <= 6.5d-65) then
        tmp = y * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z - (y * z);
	double tmp;
	if (z <= -2.9e-133) {
		tmp = t_0;
	} else if (z <= 6.5e-65) {
		tmp = y * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z - (y * z)
	tmp = 0
	if z <= -2.9e-133:
		tmp = t_0
	elif z <= 6.5e-65:
		tmp = y * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z - Float64(y * z))
	tmp = 0.0
	if (z <= -2.9e-133)
		tmp = t_0;
	elseif (z <= 6.5e-65)
		tmp = Float64(y * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z - (y * z);
	tmp = 0.0;
	if (z <= -2.9e-133)
		tmp = t_0;
	elseif (z <= 6.5e-65)
		tmp = y * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.9e-133], t$95$0, If[LessEqual[z, 6.5e-65], N[(y * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8999999999999998e-133 or 6.5e-65 < z

   1. Initial program 96.4%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if -2.8999999999999998e-133 < z < 6.5e-65

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-133}:\\ \;\;\;\;z - y \cdot z\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-65}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-98}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(z, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8e-98) (* y x) (if (<= y 4.3e-7) (fma z y z) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8e-98) {
		tmp = y * x;
	} else if (y <= 4.3e-7) {
		tmp = fma(z, y, z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8e-98)
		tmp = Float64(y * x);
	elseif (y <= 4.3e-7)
		tmp = fma(z, y, z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8e-98], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.3e-7], N[(z * y + z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.99999999999999951e-98 or 4.3000000000000001e-7 < y

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

   if -7.99999999999999951e-98 < y < 4.3000000000000001e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. neg-sub0 N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. sub0-neg N/A

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

      5. cube-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. sqr-pow N/A

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

      8. unpow-prod-down N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqr-neg N/A

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

      12. pow-prod-down N/A

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

      13. sqr-pow N/A

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

      14. metadata-eval N/A

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

      15. +-lft-identity N/A

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

      16. distribute-rgt-out N/A

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

      17. +-commutative N/A

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

      18. +-lft-identity N/A

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

      19. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{z}^{3}}{\color{blue}{{z}^{2}}}, y, z\right) \]

      20. pow-div N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{z}^{\left(3 - 2\right)}}, y, z\right) \]

      21. metadata-eval N/A

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

      22. unpow1 73.8

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

   9. Applied rewrites 73.8%

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

Alternative 6: 42.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 42.6

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

5. Applied rewrites 42.6%

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

Developer Target 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment:bezierClip from diagrams-lib-1.3.0.3"
  :precision binary64

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

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