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

Percentage Accurate: 97.8% → 100.0%

Time: 11.5s

Alternatives: 8

Speedup: 1.3×

• 21.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) + (z * (1.0d0 - y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.8% 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, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

   1. +-commutative 96.5%

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

   2. +-lft-identity 96.5%

      \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

   3. cancel-sign-sub 96.5%

      \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

   4. cancel-sign-sub 96.5%

      \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

   5. +-lft-identity 96.5%

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

   6. distribute-lft-out-- 96.5%

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

   7. *-rgt-identity 96.5%

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

   8. associate-+l- 96.5%

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

   9. distribute-rgt-out-- 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{z - y \cdot \left(z - x\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 62.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-11}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-17}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+41}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.25e-11)
   (* y x)
   (if (<= y 1.05e-17) z (if (<= y 5e+41) (* y x) (* y (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.25e-11) {
		tmp = y * x;
	} else if (y <= 1.05e-17) {
		tmp = z;
	} else if (y <= 5e+41) {
		tmp = y * x;
	} else {
		tmp = y * -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 <= (-2.25d-11)) then
        tmp = y * x
    else if (y <= 1.05d-17) then
        tmp = z
    else if (y <= 5d+41) then
        tmp = y * x
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.25e-11) {
		tmp = y * x;
	} else if (y <= 1.05e-17) {
		tmp = z;
	} else if (y <= 5e+41) {
		tmp = y * x;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.25e-11:
		tmp = y * x
	elif y <= 1.05e-17:
		tmp = z
	elif y <= 5e+41:
		tmp = y * x
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.25e-11)
		tmp = Float64(y * x);
	elseif (y <= 1.05e-17)
		tmp = z;
	elseif (y <= 5e+41)
		tmp = Float64(y * x);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.25e-11)
		tmp = y * x;
	elseif (y <= 1.05e-17)
		tmp = z;
	elseif (y <= 5e+41)
		tmp = y * x;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.25e-11], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.05e-17], z, If[LessEqual[y, 5e+41], N[(y * x), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.25e-11 or 1.04999999999999996e-17 < y < 5.00000000000000022e41

   1. Initial program 90.7%

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

      1. +-commutative 90.7%

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

      2. +-lft-identity 90.7%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      3. cancel-sign-sub 90.7%

         \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

      4. cancel-sign-sub 90.7%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      5. +-lft-identity 90.7%

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

      6. distribute-lft-out-- 90.7%

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

      7. *-rgt-identity 90.7%

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

      8. associate-+l- 90.7%

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

      9. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 60.2%

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

   if -2.25e-11 < y < 1.04999999999999996e-17

   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. +-lft-identity 100.0%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

      4. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      5. +-lft-identity 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. associate-+l- 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 77.2%

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

   6. Taylor expanded in y around 0 77.2%

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

   if 5.00000000000000022e41 < y

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

      2. +-lft-identity 97.0%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      3. cancel-sign-sub 97.0%

         \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

      4. cancel-sign-sub 97.0%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      5. +-lft-identity 97.0%

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

      6. distribute-lft-out-- 97.0%

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

      7. *-rgt-identity 97.0%

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

      8. associate-+l- 97.0%

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

      9. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

   6. Taylor expanded in x around 0 60.4%

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

      1. associate-*r* 60.4%

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

      2. neg-mul-1 60.4%

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

   8. Simplified 60.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-11}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-17}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+41}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -22000 \lor \neg \left(y \leq 0.96\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -22000.0) (not (<= y 0.96))) (* y (- x z)) (+ z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -22000.0) || !(y <= 0.96)) {
		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 <= (-22000.0d0)) .or. (.not. (y <= 0.96d0))) 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 <= -22000.0) || !(y <= 0.96)) {
		tmp = y * (x - z);
	} else {
		tmp = z + (y * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -22000.0) or not (y <= 0.96):
		tmp = y * (x - z)
	else:
		tmp = z + (y * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -22000.0) || !(y <= 0.96))
		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 <= -22000.0) || ~((y <= 0.96)))
		tmp = y * (x - z);
	else
		tmp = z + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -22000.0], N[Not[LessEqual[y, 0.96]], $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 < -22000 or 0.95999999999999996 < y

   1. Initial program 93.3%

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

      1. +-commutative 93.3%

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

      2. +-lft-identity 93.3%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      3. cancel-sign-sub 93.3%

         \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

      4. cancel-sign-sub 93.3%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      5. +-lft-identity 93.3%

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

      6. distribute-lft-out-- 93.3%

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

      7. *-rgt-identity 93.3%

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

      8. associate-+l- 93.3%

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

      9. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 99.4%

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

   if -22000 < y < 0.95999999999999996

   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. +-lft-identity 100.0%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

      4. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      5. +-lft-identity 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. associate-+l- 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. distribute-lft-neg-out 99.5%

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

      3. *-commutative 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -22000 \lor \neg \left(y \leq 0.96\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-11} \lor \neg \left(y \leq 0.00105\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.5e-11) (not (<= y 0.00105))) (* y (- x z)) (- z (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.5e-11) || !(y <= 0.00105)) {
		tmp = y * (x - z);
	} else {
		tmp = z - (z * 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 <= (-2.5d-11)) .or. (.not. (y <= 0.00105d0))) then
        tmp = y * (x - z)
    else
        tmp = z - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.5e-11) || !(y <= 0.00105)) {
		tmp = y * (x - z);
	} else {
		tmp = z - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.5e-11) or not (y <= 0.00105):
		tmp = y * (x - z)
	else:
		tmp = z - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.5e-11) || !(y <= 0.00105))
		tmp = Float64(y * Float64(x - z));
	else
		tmp = Float64(z - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.5e-11) || ~((y <= 0.00105)))
		tmp = y * (x - z);
	else
		tmp = z - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.5e-11], N[Not[LessEqual[y, 0.00105]], $MachinePrecision]], N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision], N[(z - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.50000000000000009e-11 or 0.00104999999999999994 < y

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. +-lft-identity 93.5%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      3. cancel-sign-sub 93.5%

         \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

      4. cancel-sign-sub 93.5%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      5. +-lft-identity 93.5%

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

      6. distribute-lft-out-- 93.5%

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

      7. *-rgt-identity 93.5%

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

      8. associate-+l- 93.5%

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

      9. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 99.4%

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

   if -2.50000000000000009e-11 < y < 0.00104999999999999994

   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. +-lft-identity 100.0%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

      4. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      5. +-lft-identity 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. associate-+l- 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 76.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-11} \lor \neg \left(y \leq 0.00105\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-15} \lor \neg \left(y \leq 5.5 \cdot 10^{-15}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4e-15) (not (<= y 5.5e-15))) (* y (- x z)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4e-15) || !(y <= 5.5e-15)) {
		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 <= (-4d-15)) .or. (.not. (y <= 5.5d-15))) 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 <= -4e-15) || !(y <= 5.5e-15)) {
		tmp = y * (x - z);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4e-15) or not (y <= 5.5e-15):
		tmp = y * (x - z)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4e-15) || !(y <= 5.5e-15))
		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 <= -4e-15) || ~((y <= 5.5e-15)))
		tmp = y * (x - z);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4e-15], N[Not[LessEqual[y, 5.5e-15]], $MachinePrecision]], N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if y < -4.0000000000000003e-15 or 5.5000000000000002e-15 < y

   1. Initial program 93.6%

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

      1. +-commutative 93.6%

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

      2. +-lft-identity 93.6%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      3. cancel-sign-sub 93.6%

         \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

      4. cancel-sign-sub 93.6%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      5. +-lft-identity 93.6%

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

      6. distribute-lft-out-- 93.6%

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

      7. *-rgt-identity 93.6%

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

      8. associate-+l- 93.6%

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

      9. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 98.7%

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

   if -4.0000000000000003e-15 < y < 5.5000000000000002e-15

   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. +-lft-identity 100.0%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

      4. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      5. +-lft-identity 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. associate-+l- 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 77.2%

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

   6. Taylor expanded in y around 0 77.2%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-15} \lor \neg \left(y \leq 5.5 \cdot 10^{-15}\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-12} \lor \neg \left(y \leq 7 \cdot 10^{-15}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -9.2e-12) (not (<= y 7e-15))) (* y x) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9.2e-12) || !(y <= 7e-15)) {
		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 <= (-9.2d-12)) .or. (.not. (y <= 7d-15))) 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 <= -9.2e-12) || !(y <= 7e-15)) {
		tmp = y * x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -9.2e-12) or not (y <= 7e-15):
		tmp = y * x
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -9.2e-12) || !(y <= 7e-15))
		tmp = Float64(y * x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -9.2e-12) || ~((y <= 7e-15)))
		tmp = y * x;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -9.2e-12], N[Not[LessEqual[y, 7e-15]], $MachinePrecision]], N[(y * x), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if y < -9.19999999999999957e-12 or 7.0000000000000001e-15 < y

   1. Initial program 93.6%

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

      1. +-commutative 93.6%

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

      2. +-lft-identity 93.6%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      3. cancel-sign-sub 93.6%

         \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

      4. cancel-sign-sub 93.6%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      5. +-lft-identity 93.6%

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

      6. distribute-lft-out-- 93.6%

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

      7. *-rgt-identity 93.6%

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

      8. associate-+l- 93.6%

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

      9. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 54.7%

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

   if -9.19999999999999957e-12 < y < 7.0000000000000001e-15

   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. +-lft-identity 100.0%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

      4. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

      5. +-lft-identity 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. associate-+l- 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 77.2%

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

   6. Taylor expanded in y around 0 77.2%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-12} \lor \neg \left(y \leq 7 \cdot 10^{-15}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.8% 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 96.5%

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

   1. +-commutative 96.5%

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

   2. +-lft-identity 96.5%

      \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

   3. cancel-sign-sub 96.5%

      \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

   4. cancel-sign-sub 96.5%

      \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

   5. +-lft-identity 96.5%

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

   6. distribute-lft-out-- 96.5%

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

   7. *-rgt-identity 96.5%

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

   8. associate-+l- 96.5%

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

   9. distribute-rgt-out-- 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in z around inf 63.4%

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

6. Taylor expanded in y around 0 36.3%

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

Alternative 8: 2.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 0.0)

C

double code(double x, double y, double z) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 0.0;
}

Python

def code(x, y, z):
	return 0.0

Julia

function code(x, y, z)
	return 0.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_] := 0.0

Derivation

1. Initial program 96.5%

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

   1. +-commutative 96.5%

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

   2. +-lft-identity 96.5%

      \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

   3. cancel-sign-sub 96.5%

      \[\leadsto \color{blue}{\left(0 - \left(-z\right) \cdot \left(1 - y\right)\right)} + x \cdot y \]

   4. cancel-sign-sub 96.5%

      \[\leadsto \color{blue}{\left(0 + z \cdot \left(1 - y\right)\right)} + x \cdot y \]

   5. +-lft-identity 96.5%

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

   6. distribute-lft-out-- 96.5%

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

   7. *-rgt-identity 96.5%

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

   8. associate-+l- 96.5%

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

   9. distribute-rgt-out-- 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in y around inf 65.9%

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

6. Taylor expanded in x around 0 29.8%

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

   1. associate-*r* 29.8%

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

   2. neg-mul-1 29.8%

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

8. Simplified 29.8%

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

10. Applied egg-rr 2.5%

    \[\leadsto \color{blue}{0} \cdot z \]

11. Taylor expanded in z around 0 2.5%

    \[\leadsto \color{blue}{0} \]
12. 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 2024111 
(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))))