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

Percentage Accurate: 97.6% → 100.0%

Time: 7.4s

Alternatives: 6

Speedup: 1.7×

• 21.2% 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: 97.6% 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 97.3%

   \[x \cdot y + z \cdot \left(1 - y\right) \]
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. Simplified 100.0%

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

Alternative 2: 61.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-z\right)\\ \mathbf{if}\;y \leq -2.15 \cdot 10^{+61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-14}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(z, y, z\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- z))))
   (if (<= y -2.15e+61)
     t_0
     (if (<= y -5e-14)
       (* y x)
       (if (<= y 2.95e-21) (fma z y z) (if (<= y 5.8e+14) (* y x) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = y * -z;
	double tmp;
	if (y <= -2.15e+61) {
		tmp = t_0;
	} else if (y <= -5e-14) {
		tmp = y * x;
	} else if (y <= 2.95e-21) {
		tmp = fma(z, y, z);
	} else if (y <= 5.8e+14) {
		tmp = y * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(-z))
	tmp = 0.0
	if (y <= -2.15e+61)
		tmp = t_0;
	elseif (y <= -5e-14)
		tmp = Float64(y * x);
	elseif (y <= 2.95e-21)
		tmp = fma(z, y, z);
	elseif (y <= 5.8e+14)
		tmp = Float64(y * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * (-z)), $MachinePrecision]}, If[LessEqual[y, -2.15e+61], t$95$0, If[LessEqual[y, -5e-14], N[(y * x), $MachinePrecision], If[LessEqual[y, 2.95e-21], N[(z * y + z), $MachinePrecision], If[LessEqual[y, 5.8e+14], N[(y * x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1500000000000001e61 or 5.8e14 < y

   1. Initial program 93.1%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 63.3

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

   8. Simplified 63.3%

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

   if -2.1500000000000001e61 < y < -5.0000000000000002e-14 or 2.9500000000000001e-21 < y < 5.8e14

   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 56.8

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

   5. Simplified 56.8%

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

   if -5.0000000000000002e-14 < y < 2.9500000000000001e-21

   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 76.6

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

   5. Simplified 76.6%

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

      1. *-commutative N/A

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

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

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

      3. lift-neg.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft1-in N/A

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

      6. *-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-rgt-neg-out N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 76.6

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

   7. Applied egg-rr 76.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 76.6

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

   9. Applied egg-rr 76.5%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-14}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(z, y, z\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x - z\right)\\ \mathbf{if}\;y \leq -75000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;z - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- x z))))
   (if (<= y -75000000.0) t_0 (if (<= y 0.2) (- z (* y z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x - z);
	double tmp;
	if (y <= -75000000.0) {
		tmp = t_0;
	} else if (y <= 0.2) {
		tmp = z - (y * 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 = y * (x - z)
    if (y <= (-75000000.0d0)) then
        tmp = t_0
    else if (y <= 0.2d0) then
        tmp = z - (y * 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 = y * (x - z);
	double tmp;
	if (y <= -75000000.0) {
		tmp = t_0;
	} else if (y <= 0.2) {
		tmp = z - (y * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x - z)
	tmp = 0
	if y <= -75000000.0:
		tmp = t_0
	elif y <= 0.2:
		tmp = z - (y * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x - z))
	tmp = 0.0
	if (y <= -75000000.0)
		tmp = t_0;
	elseif (y <= 0.2)
		tmp = Float64(z - Float64(y * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x - z);
	tmp = 0.0;
	if (y <= -75000000.0)
		tmp = t_0;
	elseif (y <= 0.2)
		tmp = z - (y * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -75000000.0], t$95$0, If[LessEqual[y, 0.2], N[(z - N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5e7 or 0.20000000000000001 < y

   1. Initial program 94.1%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 98.9

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

   5. Simplified 98.9%

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

   if -7.5e7 < y < 0.20000000000000001

   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 75.2

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

   5. Simplified 75.2%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -75000000:\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;z - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x - z\right)\\ \mathbf{if}\;y \leq -7.4 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(z, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- x z))))
   (if (<= y -7.4e-13) t_0 (if (<= y 3.6e-19) (fma z y z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x - z);
	double tmp;
	if (y <= -7.4e-13) {
		tmp = t_0;
	} else if (y <= 3.6e-19) {
		tmp = fma(z, y, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x - z))
	tmp = 0.0
	if (y <= -7.4e-13)
		tmp = t_0;
	elseif (y <= 3.6e-19)
		tmp = fma(z, y, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.4e-13], t$95$0, If[LessEqual[y, 3.6e-19], N[(z * y + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.39999999999999977e-13 or 3.6000000000000001e-19 < y

   1. Initial program 94.6%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 95.0

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

   5. Simplified 95.0%

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

   if -7.39999999999999977e-13 < y < 3.6000000000000001e-19

   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 76.1

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

   5. Simplified 76.1%

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

      1. *-commutative N/A

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

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

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

      3. lift-neg.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft1-in N/A

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

      6. *-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-rgt-neg-out N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 76.1

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

   7. Applied egg-rr 76.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 76.1

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

   9. Applied egg-rr 76.1%

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

Alternative 5: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-13}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(z, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.85e-13) (* y x) (if (<= y 3.3e-22) (fma z y z) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.85e-13) {
		tmp = y * x;
	} else if (y <= 3.3e-22) {
		tmp = fma(z, y, z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.85e-13)
		tmp = Float64(y * x);
	elseif (y <= 3.3e-22)
		tmp = fma(z, y, z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.85e-13], N[(y * x), $MachinePrecision], If[LessEqual[y, 3.3e-22], N[(z * y + z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.84999999999999994e-13 or 3.3000000000000001e-22 < y

   1. Initial program 94.6%

      \[x \cdot y + z \cdot \left(1 - y\right) \]
   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 44.2

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

   5. Simplified 44.2%

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

   if -1.84999999999999994e-13 < y < 3.3000000000000001e-22

   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 76.6

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

   5. Simplified 76.6%

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

      1. *-commutative N/A

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

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

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

      3. lift-neg.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft1-in N/A

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

      6. *-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-rgt-neg-out N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 76.6

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

   7. Applied egg-rr 76.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 76.6

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

   9. Applied egg-rr 76.5%

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

Alternative 6: 41.7% 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 97.3%

   \[x \cdot y + z \cdot \left(1 - y\right) \]
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 35.0

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

5. Simplified 35.0%

   \[\leadsto \color{blue}{y \cdot x} \]
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 2024207 
(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))))