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

Percentage Accurate: 97.8% → 100.0%

Time: 4.0s

Alternatives: 7

Speedup: 1.3×

• 21.1% 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: 97.8% 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, 0.1× 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 96.9%

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

   1. +-commutative 96.9%

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

   2. sub-neg 96.9%

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

   3. distribute-rgt-in 96.9%

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

   4. *-lft-identity 96.9%

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

   5. associate-+l+ 96.9%

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

   6. +-commutative 96.9%

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

   7. *-commutative 96.9%

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

   8. neg-mul-1 96.9%

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

   9. associate-*r* 96.9%

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

   10. distribute-rgt-out 100.0%

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

   11. fma-def 100.0%

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

   12. +-commutative 100.0%

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

   13. *-commutative 100.0%

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

   14. neg-mul-1 100.0%

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

   15. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 60.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-z\right)\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+96}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -55000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-16}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-54}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-73}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{-127}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-96}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-42}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+275}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- z))))
   (if (<= y -3.2e+153)
     t_0
     (if (<= y -2.1e+96)
       (* y x)
       (if (<= y -55000000000000.0)
         t_0
         (if (<= y -7.5e-16)
           (* y x)
           (if (<= y -2e-54)
             z
             (if (<= y -2.7e-73)
               (* y x)
               (if (<= y 1.62e-127)
                 z
                 (if (<= y 6.8e-96)
                   (* y x)
                   (if (<= y 1.45e-42)
                     z
                     (if (<= y 4.6e+275) (* y x) t_0))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * -z;
	double tmp;
	if (y <= -3.2e+153) {
		tmp = t_0;
	} else if (y <= -2.1e+96) {
		tmp = y * x;
	} else if (y <= -55000000000000.0) {
		tmp = t_0;
	} else if (y <= -7.5e-16) {
		tmp = y * x;
	} else if (y <= -2e-54) {
		tmp = z;
	} else if (y <= -2.7e-73) {
		tmp = y * x;
	} else if (y <= 1.62e-127) {
		tmp = z;
	} else if (y <= 6.8e-96) {
		tmp = y * x;
	} else if (y <= 1.45e-42) {
		tmp = z;
	} else if (y <= 4.6e+275) {
		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 = y * -z
    if (y <= (-3.2d+153)) then
        tmp = t_0
    else if (y <= (-2.1d+96)) then
        tmp = y * x
    else if (y <= (-55000000000000.0d0)) then
        tmp = t_0
    else if (y <= (-7.5d-16)) then
        tmp = y * x
    else if (y <= (-2d-54)) then
        tmp = z
    else if (y <= (-2.7d-73)) then
        tmp = y * x
    else if (y <= 1.62d-127) then
        tmp = z
    else if (y <= 6.8d-96) then
        tmp = y * x
    else if (y <= 1.45d-42) then
        tmp = z
    else if (y <= 4.6d+275) 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 = y * -z;
	double tmp;
	if (y <= -3.2e+153) {
		tmp = t_0;
	} else if (y <= -2.1e+96) {
		tmp = y * x;
	} else if (y <= -55000000000000.0) {
		tmp = t_0;
	} else if (y <= -7.5e-16) {
		tmp = y * x;
	} else if (y <= -2e-54) {
		tmp = z;
	} else if (y <= -2.7e-73) {
		tmp = y * x;
	} else if (y <= 1.62e-127) {
		tmp = z;
	} else if (y <= 6.8e-96) {
		tmp = y * x;
	} else if (y <= 1.45e-42) {
		tmp = z;
	} else if (y <= 4.6e+275) {
		tmp = y * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * -z
	tmp = 0
	if y <= -3.2e+153:
		tmp = t_0
	elif y <= -2.1e+96:
		tmp = y * x
	elif y <= -55000000000000.0:
		tmp = t_0
	elif y <= -7.5e-16:
		tmp = y * x
	elif y <= -2e-54:
		tmp = z
	elif y <= -2.7e-73:
		tmp = y * x
	elif y <= 1.62e-127:
		tmp = z
	elif y <= 6.8e-96:
		tmp = y * x
	elif y <= 1.45e-42:
		tmp = z
	elif y <= 4.6e+275:
		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 <= -3.2e+153)
		tmp = t_0;
	elseif (y <= -2.1e+96)
		tmp = Float64(y * x);
	elseif (y <= -55000000000000.0)
		tmp = t_0;
	elseif (y <= -7.5e-16)
		tmp = Float64(y * x);
	elseif (y <= -2e-54)
		tmp = z;
	elseif (y <= -2.7e-73)
		tmp = Float64(y * x);
	elseif (y <= 1.62e-127)
		tmp = z;
	elseif (y <= 6.8e-96)
		tmp = Float64(y * x);
	elseif (y <= 1.45e-42)
		tmp = z;
	elseif (y <= 4.6e+275)
		tmp = Float64(y * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * -z;
	tmp = 0.0;
	if (y <= -3.2e+153)
		tmp = t_0;
	elseif (y <= -2.1e+96)
		tmp = y * x;
	elseif (y <= -55000000000000.0)
		tmp = t_0;
	elseif (y <= -7.5e-16)
		tmp = y * x;
	elseif (y <= -2e-54)
		tmp = z;
	elseif (y <= -2.7e-73)
		tmp = y * x;
	elseif (y <= 1.62e-127)
		tmp = z;
	elseif (y <= 6.8e-96)
		tmp = y * x;
	elseif (y <= 1.45e-42)
		tmp = z;
	elseif (y <= 4.6e+275)
		tmp = y * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * (-z)), $MachinePrecision]}, If[LessEqual[y, -3.2e+153], t$95$0, If[LessEqual[y, -2.1e+96], N[(y * x), $MachinePrecision], If[LessEqual[y, -55000000000000.0], t$95$0, If[LessEqual[y, -7.5e-16], N[(y * x), $MachinePrecision], If[LessEqual[y, -2e-54], z, If[LessEqual[y, -2.7e-73], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.62e-127], z, If[LessEqual[y, 6.8e-96], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.45e-42], z, If[LessEqual[y, 4.6e+275], N[(y * x), $MachinePrecision], t$95$0]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2000000000000001e153 or -2.1000000000000001e96 < y < -5.5e13 or 4.60000000000000021e275 < y

   1. Initial program 93.8%

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

   2. Taylor expanded in y around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. sub-neg 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around 0 70.7%

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

      1. associate-*r* 70.7%

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

      2. neg-mul-1 70.7%

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

   7. Simplified 70.7%

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

   if -3.2000000000000001e153 < y < -2.1000000000000001e96 or -5.5e13 < y < -7.5e-16 or -2.0000000000000001e-54 < y < -2.69999999999999994e-73 or 1.61999999999999993e-127 < y < 6.8000000000000002e-96 or 1.4500000000000001e-42 < y < 4.60000000000000021e275

   1. Initial program 95.6%

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

   2. Taylor expanded in x around inf 74.5%

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

   if -7.5e-16 < y < -2.0000000000000001e-54 or -2.69999999999999994e-73 < y < 1.61999999999999993e-127 or 6.8000000000000002e-96 < y < 1.4500000000000001e-42

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 80.8%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+96}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -55000000000000:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-16}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-54}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-73}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{-127}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-96}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-42}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+275}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]

Alternative 3: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x - z\right)\\ \mathbf{if}\;y \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-51}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-127}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-92}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-42}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- x z))))
   (if (<= y -2e-15)
     t_0
     (if (<= y -1.8e-51)
       z
       (if (<= y -1.9e-73)
         t_0
         (if (<= y 1.55e-127)
           z
           (if (<= y 3.5e-92) (* y x) (if (<= y 8.5e-42) z t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x - z);
	double tmp;
	if (y <= -2e-15) {
		tmp = t_0;
	} else if (y <= -1.8e-51) {
		tmp = z;
	} else if (y <= -1.9e-73) {
		tmp = t_0;
	} else if (y <= 1.55e-127) {
		tmp = z;
	} else if (y <= 3.5e-92) {
		tmp = y * x;
	} else if (y <= 8.5e-42) {
		tmp = 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 <= (-2d-15)) then
        tmp = t_0
    else if (y <= (-1.8d-51)) then
        tmp = z
    else if (y <= (-1.9d-73)) then
        tmp = t_0
    else if (y <= 1.55d-127) then
        tmp = z
    else if (y <= 3.5d-92) then
        tmp = y * x
    else if (y <= 8.5d-42) then
        tmp = 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 <= -2e-15) {
		tmp = t_0;
	} else if (y <= -1.8e-51) {
		tmp = z;
	} else if (y <= -1.9e-73) {
		tmp = t_0;
	} else if (y <= 1.55e-127) {
		tmp = z;
	} else if (y <= 3.5e-92) {
		tmp = y * x;
	} else if (y <= 8.5e-42) {
		tmp = z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x - z)
	tmp = 0
	if y <= -2e-15:
		tmp = t_0
	elif y <= -1.8e-51:
		tmp = z
	elif y <= -1.9e-73:
		tmp = t_0
	elif y <= 1.55e-127:
		tmp = z
	elif y <= 3.5e-92:
		tmp = y * x
	elif y <= 8.5e-42:
		tmp = 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 <= -2e-15)
		tmp = t_0;
	elseif (y <= -1.8e-51)
		tmp = z;
	elseif (y <= -1.9e-73)
		tmp = t_0;
	elseif (y <= 1.55e-127)
		tmp = z;
	elseif (y <= 3.5e-92)
		tmp = Float64(y * x);
	elseif (y <= 8.5e-42)
		tmp = 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 <= -2e-15)
		tmp = t_0;
	elseif (y <= -1.8e-51)
		tmp = z;
	elseif (y <= -1.9e-73)
		tmp = t_0;
	elseif (y <= 1.55e-127)
		tmp = z;
	elseif (y <= 3.5e-92)
		tmp = y * x;
	elseif (y <= 8.5e-42)
		tmp = 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, -2e-15], t$95$0, If[LessEqual[y, -1.8e-51], z, If[LessEqual[y, -1.9e-73], t$95$0, If[LessEqual[y, 1.55e-127], z, If[LessEqual[y, 3.5e-92], N[(y * x), $MachinePrecision], If[LessEqual[y, 8.5e-42], z, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0000000000000002e-15 or -1.8e-51 < y < -1.9000000000000001e-73 or 8.4999999999999996e-42 < y

   1. Initial program 94.7%

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

   2. Taylor expanded in y around inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. +-commutative 97.3%

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

      3. sub-neg 97.3%

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

   4. Simplified 97.3%

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

   if -2.0000000000000002e-15 < y < -1.8e-51 or -1.9000000000000001e-73 < y < 1.55e-127 or 3.5e-92 < y < 8.4999999999999996e-42

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 80.8%

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

   if 1.55e-127 < y < 3.5e-92

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-15}:\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-51}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-73}:\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-127}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-92}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-42}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - z\right)\\ \end{array} \]

Alternative 4: 60.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-53}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-75}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{-127}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-96}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-42}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.6e-15)
   (* y x)
   (if (<= y -4.2e-53)
     z
     (if (<= y -3.3e-75)
       (* y x)
       (if (<= y 1.62e-127)
         z
         (if (<= y 9e-96) (* y x) (if (<= y 1.3e-42) z (* y x))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e-15) {
		tmp = y * x;
	} else if (y <= -4.2e-53) {
		tmp = z;
	} else if (y <= -3.3e-75) {
		tmp = y * x;
	} else if (y <= 1.62e-127) {
		tmp = z;
	} else if (y <= 9e-96) {
		tmp = y * x;
	} else if (y <= 1.3e-42) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.6d-15)) then
        tmp = y * x
    else if (y <= (-4.2d-53)) then
        tmp = z
    else if (y <= (-3.3d-75)) then
        tmp = y * x
    else if (y <= 1.62d-127) then
        tmp = z
    else if (y <= 9d-96) then
        tmp = y * x
    else if (y <= 1.3d-42) then
        tmp = z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e-15) {
		tmp = y * x;
	} else if (y <= -4.2e-53) {
		tmp = z;
	} else if (y <= -3.3e-75) {
		tmp = y * x;
	} else if (y <= 1.62e-127) {
		tmp = z;
	} else if (y <= 9e-96) {
		tmp = y * x;
	} else if (y <= 1.3e-42) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.6e-15:
		tmp = y * x
	elif y <= -4.2e-53:
		tmp = z
	elif y <= -3.3e-75:
		tmp = y * x
	elif y <= 1.62e-127:
		tmp = z
	elif y <= 9e-96:
		tmp = y * x
	elif y <= 1.3e-42:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.6e-15)
		tmp = Float64(y * x);
	elseif (y <= -4.2e-53)
		tmp = z;
	elseif (y <= -3.3e-75)
		tmp = Float64(y * x);
	elseif (y <= 1.62e-127)
		tmp = z;
	elseif (y <= 9e-96)
		tmp = Float64(y * x);
	elseif (y <= 1.3e-42)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.6e-15)
		tmp = y * x;
	elseif (y <= -4.2e-53)
		tmp = z;
	elseif (y <= -3.3e-75)
		tmp = y * x;
	elseif (y <= 1.62e-127)
		tmp = z;
	elseif (y <= 9e-96)
		tmp = y * x;
	elseif (y <= 1.3e-42)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.6e-15], N[(y * x), $MachinePrecision], If[LessEqual[y, -4.2e-53], z, If[LessEqual[y, -3.3e-75], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.62e-127], z, If[LessEqual[y, 9e-96], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.3e-42], z, N[(y * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e-15 or -4.19999999999999955e-53 < y < -3.3e-75 or 1.61999999999999993e-127 < y < 9e-96 or 1.3e-42 < y

   1. Initial program 94.9%

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

   2. Taylor expanded in x around inf 58.5%

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

   if -1.6e-15 < y < -4.19999999999999955e-53 or -3.3e-75 < y < 1.61999999999999993e-127 or 9e-96 < y < 1.3e-42

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 80.8%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-53}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-75}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{-127}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-96}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-42}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -18000000000000 \lor \neg \left(y \leq 2 \cdot 10^{-41}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -18000000000000.0) (not (<= y 2e-41)))
   (* y (- x z))
   (+ z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -18000000000000.0) || !(y <= 2e-41)) {
		tmp = y * (x - z);
	} else {
		tmp = z + (y * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-18000000000000.0d0)) .or. (.not. (y <= 2d-41))) then
        tmp = y * (x - z)
    else
        tmp = z + (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -18000000000000.0) || !(y <= 2e-41)) {
		tmp = y * (x - z);
	} else {
		tmp = z + (y * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -18000000000000.0) or not (y <= 2e-41):
		tmp = y * (x - z)
	else:
		tmp = z + (y * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -18000000000000.0) || !(y <= 2e-41))
		tmp = Float64(y * Float64(x - z));
	else
		tmp = Float64(z + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -18000000000000.0) || ~((y <= 2e-41)))
		tmp = y * (x - z);
	else
		tmp = z + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -18000000000000.0], N[Not[LessEqual[y, 2e-41]], $MachinePrecision]], N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision], N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e13 or 2.00000000000000001e-41 < y

   1. Initial program 94.0%

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

   2. Taylor expanded in y around inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. +-commutative 99.0%

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

      3. sub-neg 99.0%

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

   4. Simplified 99.0%

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

   if -1.8e13 < y < 2.00000000000000001e-41

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. neg-mul-1 100.0%

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

      9. associate-*r* 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -18000000000000 \lor \neg \left(y \leq 2 \cdot 10^{-41}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \]

Alternative 6: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

   1. +-commutative 96.9%

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

   2. sub-neg 96.9%

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

   3. distribute-rgt-in 96.9%

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

   4. *-lft-identity 96.9%

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

   5. associate-+l+ 96.9%

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

   6. +-commutative 96.9%

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

   7. *-commutative 96.9%

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

   8. neg-mul-1 96.9%

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

   9. associate-*r* 96.9%

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

   10. distribute-rgt-out 100.0%

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

   11. fma-def 100.0%

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

   12. +-commutative 100.0%

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

   13. *-commutative 100.0%

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

   14. neg-mul-1 100.0%

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

   15. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around 0 100.0%

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

5. Final simplification 100.0%

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

Alternative 7: 36.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 96.9%

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

2. Taylor expanded in y around 0 34.4%

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

3. Final simplification 34.4%

   \[\leadsto z \]

Developer target: 100.0% accurate, 1.3× 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 2023240 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment:bezierClip from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (- z (* (- z x) y))

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