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

Percentage Accurate: 97.8% → 100.0%

Time: 5.0s

Alternatives: 8

Speedup: 1.3×

• 20.7% 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 97.6%

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

   1. +-commutative 97.6%

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

   2. sub-neg 97.6%

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

   3. distribute-rgt-in 97.6%

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

   4. *-lft-identity 97.6%

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

   5. associate-+l+ 97.6%

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

   6. +-commutative 97.6%

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

   7. *-commutative 97.6%

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

   8. neg-mul-1 97.6%

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

   9. associate-*r* 97.6%

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

   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: 61.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-z\right)\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+241}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{+200}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-30}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+161}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+285}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- z))))
   (if (<= y -5.8e+241)
     t_0
     (if (<= y -7.5e+200)
       (* y x)
       (if (<= y -1.0)
         t_0
         (if (<= y 2.75e-30)
           z
           (if (<= y 2e+161) (* y x) (if (<= y 2.15e+285) t_0 (* y x)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * -z;
	double tmp;
	if (y <= -5.8e+241) {
		tmp = t_0;
	} else if (y <= -7.5e+200) {
		tmp = y * x;
	} else if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 2.75e-30) {
		tmp = z;
	} else if (y <= 2e+161) {
		tmp = y * x;
	} else if (y <= 2.15e+285) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y * -z
    if (y <= (-5.8d+241)) then
        tmp = t_0
    else if (y <= (-7.5d+200)) then
        tmp = y * x
    else if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 2.75d-30) then
        tmp = z
    else if (y <= 2d+161) then
        tmp = y * x
    else if (y <= 2.15d+285) then
        tmp = t_0
    else
        tmp = y * x
    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 <= -5.8e+241) {
		tmp = t_0;
	} else if (y <= -7.5e+200) {
		tmp = y * x;
	} else if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 2.75e-30) {
		tmp = z;
	} else if (y <= 2e+161) {
		tmp = y * x;
	} else if (y <= 2.15e+285) {
		tmp = t_0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * -z
	tmp = 0
	if y <= -5.8e+241:
		tmp = t_0
	elif y <= -7.5e+200:
		tmp = y * x
	elif y <= -1.0:
		tmp = t_0
	elif y <= 2.75e-30:
		tmp = z
	elif y <= 2e+161:
		tmp = y * x
	elif y <= 2.15e+285:
		tmp = t_0
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(-z))
	tmp = 0.0
	if (y <= -5.8e+241)
		tmp = t_0;
	elseif (y <= -7.5e+200)
		tmp = Float64(y * x);
	elseif (y <= -1.0)
		tmp = t_0;
	elseif (y <= 2.75e-30)
		tmp = z;
	elseif (y <= 2e+161)
		tmp = Float64(y * x);
	elseif (y <= 2.15e+285)
		tmp = t_0;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * -z;
	tmp = 0.0;
	if (y <= -5.8e+241)
		tmp = t_0;
	elseif (y <= -7.5e+200)
		tmp = y * x;
	elseif (y <= -1.0)
		tmp = t_0;
	elseif (y <= 2.75e-30)
		tmp = z;
	elseif (y <= 2e+161)
		tmp = y * x;
	elseif (y <= 2.15e+285)
		tmp = t_0;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * (-z)), $MachinePrecision]}, If[LessEqual[y, -5.8e+241], t$95$0, If[LessEqual[y, -7.5e+200], N[(y * x), $MachinePrecision], If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 2.75e-30], z, If[LessEqual[y, 2e+161], N[(y * x), $MachinePrecision], If[LessEqual[y, 2.15e+285], t$95$0, N[(y * x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.8000000000000004e241 or -7.50000000000000062e200 < y < -1 or 2.0000000000000001e161 < y < 2.15e285

   1. Initial program 95.8%

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

   2. Taylor expanded in y around inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. sub-neg 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in x around 0 65.9%

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

      1. associate-*r* 65.9%

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

      2. neg-mul-1 65.9%

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

   7. Simplified 65.9%

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

   if -5.8000000000000004e241 < y < -7.50000000000000062e200 or 2.74999999999999988e-30 < y < 2.0000000000000001e161 or 2.15e285 < y

   1. Initial program 94.9%

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

   2. Taylor expanded in x around inf 67.4%

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

   if -1 < y < 2.74999999999999988e-30

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 74.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+241}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{+200}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-30}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+161}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+285}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-21} \lor \neg \left(y \leq 1.2 \cdot 10^{-31}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.3e-21) (not (<= y 1.2e-31))) (* y (- x z)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.3e-21) || !(y <= 1.2e-31)) {
		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 <= (-3.3d-21)) .or. (.not. (y <= 1.2d-31))) 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 <= -3.3e-21) || !(y <= 1.2e-31)) {
		tmp = y * (x - z);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.3e-21) or not (y <= 1.2e-31):
		tmp = y * (x - z)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.3e-21) || !(y <= 1.2e-31))
		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 <= -3.3e-21) || ~((y <= 1.2e-31)))
		tmp = y * (x - z);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.3e-21], N[Not[LessEqual[y, 1.2e-31]], $MachinePrecision]], N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if y < -3.30000000000000009e-21 or 1.2e-31 < y

   1. Initial program 95.5%

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

   2. Taylor expanded in y around inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. +-commutative 98.2%

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

      3. sub-neg 98.2%

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

   4. Simplified 98.2%

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

   if -3.30000000000000009e-21 < y < 1.2e-31

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 76.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-21} \lor \neg \left(y \leq 1.2 \cdot 10^{-31}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-21} \lor \neg \left(y \leq 10^{-30}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.4e-21) (not (<= y 1e-30))) (* y (- x z)) (* z (- 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.4e-21) || !(y <= 1e-30)) {
		tmp = y * (x - z);
	} else {
		tmp = z * (1.0 - y);
	}
	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 <= (-5.4d-21)) .or. (.not. (y <= 1d-30))) then
        tmp = y * (x - z)
    else
        tmp = z * (1.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.4e-21) || !(y <= 1e-30)) {
		tmp = y * (x - z);
	} else {
		tmp = z * (1.0 - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.4e-21) or not (y <= 1e-30):
		tmp = y * (x - z)
	else:
		tmp = z * (1.0 - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.4e-21) || !(y <= 1e-30))
		tmp = Float64(y * Float64(x - z));
	else
		tmp = Float64(z * Float64(1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5.4e-21) || ~((y <= 1e-30)))
		tmp = y * (x - z);
	else
		tmp = z * (1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.4e-21], N[Not[LessEqual[y, 1e-30]], $MachinePrecision]], N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision], N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4000000000000002e-21 or 1e-30 < y

   1. Initial program 95.5%

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

   2. Taylor expanded in y around inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. +-commutative 98.2%

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

      3. sub-neg 98.2%

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

   4. Simplified 98.2%

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

   if -5.4000000000000002e-21 < y < 1e-30

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 76.0%

      \[\leadsto \color{blue}{z \cdot \left(1 - y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-21} \lor \neg \left(y \leq 10^{-30}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - y\right)\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.35 \cdot 10^{-6}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.35e-6))) (* y (- x z)) (+ z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.35e-6)) {
		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 <= (-1.0d0)) .or. (.not. (y <= 1.35d-6))) 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 <= -1.0) || !(y <= 1.35e-6)) {
		tmp = y * (x - z);
	} else {
		tmp = z + (y * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.35e-6):
		tmp = y * (x - z)
	else:
		tmp = z + (y * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.35e-6))
		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 <= -1.0) || ~((y <= 1.35e-6)))
		tmp = y * (x - z);
	else
		tmp = z + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.35e-6]], $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 or 1.34999999999999999e-6 < y

   1. Initial program 95.2%

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

   2. Taylor expanded in y around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. sub-neg 99.6%

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

   4. Simplified 99.6%

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

   if -1 < y < 1.34999999999999999e-6

   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. Step-by-step derivation

      1. flip-- 65.4%

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

      2. associate-*r/ 64.9%

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

      3. difference-of-squares 65.0%

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

      4. add-sqr-sqrt 31.0%

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

      5. sqrt-unprod 64.7%

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

      6. sqr-neg 64.7%

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

      7. sqrt-unprod 33.7%

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

      8. add-sqr-sqrt 64.6%

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

      9. sub-neg 64.6%

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

      10. pow2 64.6%

          \[\leadsto \frac{y \cdot \color{blue}{{\left(x - z\right)}^{2}}}{x + z} + z \]

      11. sub-neg 64.6%

          \[\leadsto \frac{y \cdot {\color{blue}{\left(x + \left(-z\right)\right)}}^{2}}{x + z} + z \]

      12. add-sqr-sqrt 33.7%

          \[\leadsto \frac{y \cdot {\left(x + \color{blue}{\sqrt{-z} \cdot \sqrt{-z}}\right)}^{2}}{x + z} + z \]

      13. sqrt-unprod 64.6%

          \[\leadsto \frac{y \cdot {\left(x + \color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}\right)}^{2}}{x + z} + z \]

      14. sqr-neg 64.6%

          \[\leadsto \frac{y \cdot {\left(x + \sqrt{\color{blue}{z \cdot z}}\right)}^{2}}{x + z} + z \]

      15. sqrt-unprod 31.0%

          \[\leadsto \frac{y \cdot {\left(x + \color{blue}{\sqrt{z} \cdot \sqrt{z}}\right)}^{2}}{x + z} + z \]

      16. add-sqr-sqrt 64.6%

          \[\leadsto \frac{y \cdot {\left(x + \color{blue}{z}\right)}^{2}}{x + z} + z \]

   6. Applied egg-rr 64.6%

      \[\leadsto \color{blue}{\frac{y \cdot {\left(x + z\right)}^{2}}{x + z}} + z \]
   7. Step-by-step derivation

      1. associate-/l* 65.2%

         \[\leadsto \color{blue}{\frac{y}{\frac{x + z}{{\left(x + z\right)}^{2}}}} + z \]

      2. unpow2 65.2%

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

      3. associate-/r* 99.3%

         \[\leadsto \frac{y}{\color{blue}{\frac{\frac{x + z}{x + z}}{x + z}}} + z \]

      4. *-inverses 99.3%

         \[\leadsto \frac{y}{\frac{\color{blue}{1}}{x + z}} + z \]

      5. +-commutative 99.3%

         \[\leadsto \frac{y}{\frac{1}{\color{blue}{z + x}}} + z \]

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{y}{\frac{1}{z + x}}} + z \]

   9. Taylor expanded in z around 0 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.35 \cdot 10^{-6}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \]

Alternative 6: 62.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-21}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{-30}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.7e-21) (* y x) (if (<= y 3.05e-30) z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.7e-21) {
		tmp = y * x;
	} else if (y <= 3.05e-30) {
		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 <= (-1.7d-21)) then
        tmp = y * x
    else if (y <= 3.05d-30) 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 <= -1.7e-21) {
		tmp = y * x;
	} else if (y <= 3.05e-30) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.7e-21:
		tmp = y * x
	elif y <= 3.05e-30:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.7e-21)
		tmp = Float64(y * x);
	elseif (y <= 3.05e-30)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.7e-21)
		tmp = y * x;
	elseif (y <= 3.05e-30)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.7e-21], N[(y * x), $MachinePrecision], If[LessEqual[y, 3.05e-30], z, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e-21 or 3.0499999999999999e-30 < y

   1. Initial program 95.5%

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

   2. Taylor expanded in x around inf 51.4%

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

   if -1.7e-21 < y < 3.0499999999999999e-30

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 76.0%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-21}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{-30}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

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 97.6%

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

   1. +-commutative 97.6%

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

   2. sub-neg 97.6%

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

   3. distribute-rgt-in 97.6%

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

   4. *-lft-identity 97.6%

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

   5. associate-+l+ 97.6%

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

   6. +-commutative 97.6%

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

   7. *-commutative 97.6%

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

   8. neg-mul-1 97.6%

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

   9. associate-*r* 97.6%

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

   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 8: 36.1% 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 97.6%

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

2. Taylor expanded in y around 0 38.4%

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

3. Final simplification 38.4%

   \[\leadsto z \]

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 2023213 
(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))))