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

Percentage Accurate: 98.1% → 100.0%

Time: 6.1s

Alternatives: 8

Speedup: 1.3×

• 20.3% 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, 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 97.2%

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

   1. distribute-lft-out-- 97.2%

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

   2. *-rgt-identity 97.2%

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

   3. cancel-sign-sub-inv 97.2%

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

   4. +-commutative 97.2%

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

   5. associate-+r+ 97.3%

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

   6. distribute-rgt-out 100.0%

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

   7. +-commutative 100.0%

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

   8. fma-define 100.0%

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

   9. +-commutative 100.0%

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

   10. 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. Add Preprocessing
5. Add Preprocessing

Alternative 2: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x - z\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-78}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-40}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- x z))))
   (if (<= y -2.2e-54)
     t_0
     (if (<= y 1.75e-78)
       z
       (if (<= y 9.5e-40) (* y x) (if (<= y 4.8e-6) (* z (- 1.0 y)) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x - z);
	double tmp;
	if (y <= -2.2e-54) {
		tmp = t_0;
	} else if (y <= 1.75e-78) {
		tmp = z;
	} else if (y <= 9.5e-40) {
		tmp = y * x;
	} else if (y <= 4.8e-6) {
		tmp = z * (1.0 - y);
	} 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 <= (-2.2d-54)) then
        tmp = t_0
    else if (y <= 1.75d-78) then
        tmp = z
    else if (y <= 9.5d-40) then
        tmp = y * x
    else if (y <= 4.8d-6) then
        tmp = z * (1.0d0 - y)
    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 <= -2.2e-54) {
		tmp = t_0;
	} else if (y <= 1.75e-78) {
		tmp = z;
	} else if (y <= 9.5e-40) {
		tmp = y * x;
	} else if (y <= 4.8e-6) {
		tmp = z * (1.0 - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x - z)
	tmp = 0
	if y <= -2.2e-54:
		tmp = t_0
	elif y <= 1.75e-78:
		tmp = z
	elif y <= 9.5e-40:
		tmp = y * x
	elif y <= 4.8e-6:
		tmp = z * (1.0 - y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x - z))
	tmp = 0.0
	if (y <= -2.2e-54)
		tmp = t_0;
	elseif (y <= 1.75e-78)
		tmp = z;
	elseif (y <= 9.5e-40)
		tmp = Float64(y * x);
	elseif (y <= 4.8e-6)
		tmp = Float64(z * Float64(1.0 - y));
	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 <= -2.2e-54)
		tmp = t_0;
	elseif (y <= 1.75e-78)
		tmp = z;
	elseif (y <= 9.5e-40)
		tmp = y * x;
	elseif (y <= 4.8e-6)
		tmp = z * (1.0 - y);
	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, -2.2e-54], t$95$0, If[LessEqual[y, 1.75e-78], z, If[LessEqual[y, 9.5e-40], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.8e-6], N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.2e-54 or 4.7999999999999998e-6 < y

   1. Initial program 95.3%

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

      1. distribute-lft-out-- 95.3%

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

      2. *-rgt-identity 95.3%

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

      3. cancel-sign-sub-inv 95.3%

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

      4. +-commutative 95.3%

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

      5. associate-+r+ 95.3%

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

      6. distribute-rgt-out 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. +-commutative 100.0%

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

      10. 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. Add Preprocessing

   5. Taylor expanded in y around inf 96.6%

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

   if -2.2e-54 < y < 1.75e-78

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 85.0%

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

   if 1.75e-78 < y < 9.5000000000000006e-40

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if 9.5000000000000006e-40 < y < 4.7999999999999998e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 78.9%

      \[\leadsto \color{blue}{z \cdot \left(1 - y\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 61.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-z\right)\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+168}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -2.02 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-35}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- z))))
   (if (<= y -4.5e+168)
     (* y x)
     (if (<= y -2.02e+62)
       t_0
       (if (<= y -1.45e-35) (* y x) (if (<= y 1.0) z t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = y * -z;
	double tmp;
	if (y <= -4.5e+168) {
		tmp = y * x;
	} else if (y <= -2.02e+62) {
		tmp = t_0;
	} else if (y <= -1.45e-35) {
		tmp = y * x;
	} else if (y <= 1.0) {
		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 * -z
    if (y <= (-4.5d+168)) then
        tmp = y * x
    else if (y <= (-2.02d+62)) then
        tmp = t_0
    else if (y <= (-1.45d-35)) then
        tmp = y * x
    else if (y <= 1.0d0) 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 * -z;
	double tmp;
	if (y <= -4.5e+168) {
		tmp = y * x;
	} else if (y <= -2.02e+62) {
		tmp = t_0;
	} else if (y <= -1.45e-35) {
		tmp = y * x;
	} else if (y <= 1.0) {
		tmp = z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * -z
	tmp = 0
	if y <= -4.5e+168:
		tmp = y * x
	elif y <= -2.02e+62:
		tmp = t_0
	elif y <= -1.45e-35:
		tmp = y * x
	elif y <= 1.0:
		tmp = z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(-z))
	tmp = 0.0
	if (y <= -4.5e+168)
		tmp = Float64(y * x);
	elseif (y <= -2.02e+62)
		tmp = t_0;
	elseif (y <= -1.45e-35)
		tmp = Float64(y * x);
	elseif (y <= 1.0)
		tmp = z;
	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 <= -4.5e+168)
		tmp = y * x;
	elseif (y <= -2.02e+62)
		tmp = t_0;
	elseif (y <= -1.45e-35)
		tmp = y * x;
	elseif (y <= 1.0)
		tmp = z;
	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, -4.5e+168], N[(y * x), $MachinePrecision], If[LessEqual[y, -2.02e+62], t$95$0, If[LessEqual[y, -1.45e-35], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.0], z, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.50000000000000012e168 or -2.0200000000000001e62 < y < -1.4500000000000001e-35

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf 61.5%

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

      1. *-commutative 61.5%

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

   5. Simplified 61.5%

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

   if -4.50000000000000012e168 < y < -2.0200000000000001e62 or 1 < y

   1. Initial program 93.7%

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

      1. distribute-lft-out-- 93.7%

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

      2. *-rgt-identity 93.7%

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

      3. cancel-sign-sub-inv 93.7%

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

      4. +-commutative 93.7%

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

      5. associate-+r+ 93.7%

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

      6. distribute-rgt-out 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. +-commutative 100.0%

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

      10. 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. Add Preprocessing

   5. Taylor expanded in y around inf 99.5%

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

   6. Taylor expanded in x around 0 62.9%

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

      1. mul-1-neg 62.9%

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

      2. *-commutative 62.9%

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

      3. distribute-rgt-neg-in 62.9%

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

   8. Simplified 62.9%

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

   if -1.4500000000000001e-35 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.3%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+168}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -2.02 \cdot 10^{+62}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-35}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5500 \lor \neg \left(y \leq 1.26 \cdot 10^{-5}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5500.0) (not (<= y 1.26e-5))) (* y (- x z)) (+ z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5500.0) || !(y <= 1.26e-5)) {
		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 <= (-5500.0d0)) .or. (.not. (y <= 1.26d-5))) 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 <= -5500.0) || !(y <= 1.26e-5)) {
		tmp = y * (x - z);
	} else {
		tmp = z + (y * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5500.0) or not (y <= 1.26e-5):
		tmp = y * (x - z)
	else:
		tmp = z + (y * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5500.0) || !(y <= 1.26e-5))
		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 <= -5500.0) || ~((y <= 1.26e-5)))
		tmp = y * (x - z);
	else
		tmp = z + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5500.0], N[Not[LessEqual[y, 1.26e-5]], $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 < -5500 or 1.25999999999999996e-5 < y

   1. Initial program 94.7%

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

      1. distribute-lft-out-- 94.7%

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

      2. *-rgt-identity 94.7%

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

      3. cancel-sign-sub-inv 94.7%

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

      4. +-commutative 94.7%

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

      5. associate-+r+ 94.7%

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

      6. distribute-rgt-out 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. +-commutative 100.0%

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

      10. 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. Add Preprocessing

   5. Taylor expanded in y around inf 99.7%

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

   if -5500 < y < 1.25999999999999996e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5500 \lor \neg \left(y \leq 1.26 \cdot 10^{-5}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-56} \lor \neg \left(y \leq 1.32 \cdot 10^{-76}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8.5e-56) (not (<= y 1.32e-76))) (* y (- x z)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8.5e-56) || !(y <= 1.32e-76)) {
		tmp = y * (x - z);
	} else {
		tmp = z;
	}
	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 <= (-8.5d-56)) .or. (.not. (y <= 1.32d-76))) then
        tmp = y * (x - z)
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8.5e-56) || !(y <= 1.32e-76)) {
		tmp = y * (x - z);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -8.5e-56) or not (y <= 1.32e-76):
		tmp = y * (x - z)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8.5e-56) || !(y <= 1.32e-76))
		tmp = Float64(y * Float64(x - z));
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -8.5e-56) || ~((y <= 1.32e-76)))
		tmp = y * (x - z);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8.5e-56], N[Not[LessEqual[y, 1.32e-76]], $MachinePrecision]], N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if y < -8.49999999999999932e-56 or 1.31999999999999996e-76 < y

   1. Initial program 95.7%

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

      1. distribute-lft-out-- 95.7%

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

      2. *-rgt-identity 95.7%

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

      3. cancel-sign-sub-inv 95.7%

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

      4. +-commutative 95.7%

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

      5. associate-+r+ 95.7%

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

      6. distribute-rgt-out 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. +-commutative 100.0%

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

      10. 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. Add Preprocessing

   5. Taylor expanded in y around inf 91.5%

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

   if -8.49999999999999932e-56 < y < 1.31999999999999996e-76

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 85.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-56} \lor \neg \left(y \leq 1.32 \cdot 10^{-76}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-31} \lor \neg \left(y \leq 1.2 \cdot 10^{-76}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.7e-31) (not (<= y 1.2e-76))) (* y x) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.7e-31) || !(y <= 1.2e-76)) {
		tmp = y * x;
	} else {
		tmp = z;
	}
	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 <= (-3.7d-31)) .or. (.not. (y <= 1.2d-76))) then
        tmp = y * x
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.7e-31) || !(y <= 1.2e-76)) {
		tmp = y * x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.7e-31) or not (y <= 1.2e-76):
		tmp = y * x
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.7e-31) || !(y <= 1.2e-76))
		tmp = Float64(y * x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.7e-31) || ~((y <= 1.2e-76)))
		tmp = y * x;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.7e-31], N[Not[LessEqual[y, 1.2e-76]], $MachinePrecision]], N[(y * x), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if y < -3.6999999999999998e-31 or 1.20000000000000007e-76 < y

   1. Initial program 95.6%

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

   3. Taylor expanded in x around inf 51.9%

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

      1. *-commutative 51.9%

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

   5. Simplified 51.9%

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

   if -3.6999999999999998e-31 < y < 1.20000000000000007e-76

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 83.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-31} \lor \neg \left(y \leq 1.2 \cdot 10^{-76}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 97.2%

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

   1. distribute-lft-out-- 97.2%

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

   2. *-rgt-identity 97.2%

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

   3. cancel-sign-sub-inv 97.2%

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

   4. +-commutative 97.2%

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

   5. associate-+r+ 97.3%

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

   6. distribute-rgt-out 100.0%

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

   7. +-commutative 100.0%

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

   8. fma-define 100.0%

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

   9. +-commutative 100.0%

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

   10. 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. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 8: 37.0% 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 97.2%

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

3. Taylor expanded in y around 0 36.7%

   \[\leadsto \color{blue}{z} \]
4. Add Preprocessing

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 2024097 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment:bezierClip from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (- z (* (- z x) y))

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