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

Percentage Accurate: 97.8% → 100.0%

Time: 5.7s

Alternatives: 7

Speedup: 1.3×

• 22.0% 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

\[\begin{array}{l} \\ \mathsf{fma}\left(z - x, 0 - y, z\right) \end{array} \]

FPCore

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

C

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

Julia

function code(x, y, z)
	return fma(Float64(z - x), Float64(0.0 - y), z)
end

Wolfram

code[x_, y_, z_] := N[(N[(z - x), $MachinePrecision] * N[(0.0 - y), $MachinePrecision] + z), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

   1. +-commutative N/A

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

   2. distribute-rgt-out-- N/A

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

   3. *-lft-identity N/A

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

   4. associate-+l- N/A

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

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \color{blue}{\left(y \cdot z - x \cdot y\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \left(y \cdot z - y \cdot \color{blue}{x}\right)\right) \]

   7. distribute-lft-out-- N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \left(y \cdot \color{blue}{\left(z - x\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(y, \color{blue}{\left(z - x\right)}\right)\right) \]

   9. --lowering--.f64 100.0%

      \[\leadsto \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(z, \color{blue}{x}\right)\right)\right) \]

3. Simplified 100.0%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. fma-define N/A

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

   6. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(z - x\right), \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, z\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(z, x\right), \left(\mathsf{neg}\left(\color{blue}{y}\right)\right), z\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(z, x\right), \left(0 - \color{blue}{y}\right), z\right) \]

   9. --lowering--.f64 100.0%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(z, x\right), \mathsf{\_.f64}\left(0, \color{blue}{y}\right), z\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z - x, 0 - y, z\right)} \]
7. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(z, x\right), \left(\mathsf{neg}\left(y\right)\right), z\right) \]

   2. neg-lowering-neg.f64 100.0%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(z, x\right), \mathsf{neg.f64}\left(y\right), z\right) \]

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(z - x, 0 - y, z\right) \]
10. Add Preprocessing

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x - z\right)\\ \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.00095:\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- x z))))
   (if (<= y -2.4) t_0 (if (<= y 0.00095) (+ z (* x y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x - z);
	double tmp;
	if (y <= -2.4) {
		tmp = t_0;
	} else if (y <= 0.00095) {
		tmp = z + (x * 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 <= (-2.4d0)) then
        tmp = t_0
    else if (y <= 0.00095d0) then
        tmp = z + (x * 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 <= -2.4) {
		tmp = t_0;
	} else if (y <= 0.00095) {
		tmp = z + (x * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x - z)
	tmp = 0
	if y <= -2.4:
		tmp = t_0
	elif y <= 0.00095:
		tmp = z + (x * 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 <= -2.4)
		tmp = t_0;
	elseif (y <= 0.00095)
		tmp = Float64(z + Float64(x * 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 <= -2.4)
		tmp = t_0;
	elseif (y <= 0.00095)
		tmp = z + (x * 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, -2.4], t$95$0, If[LessEqual[y, 0.00095], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.39999999999999991 or 9.49999999999999998e-4 < y

   1. Initial program 97.6%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot \left(z - x\right)\right)}\right) \]

      7. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(-1 \cdot z - \color{blue}{-1 \cdot x}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot x\right)\right)}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(-1 \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(-1 \cdot z + x\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{-1 \cdot z}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(z\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(x - \color{blue}{z}\right)\right) \]

      14. --lowering--.f64 98.4%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(x, \color{blue}{z}\right)\right) \]

   5. Simplified 98.4%

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

   if -2.39999999999999991 < y < 9.49999999999999998e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{z}\right) \]
   4. Step-by-step derivation

   5. Simplified 99.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{elif}\;y \leq 0.00095:\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x - z\right)\\ \mathbf{if}\;y \leq -2.06 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;z \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- x z))))
   (if (<= y -2.06e-10) t_0 (if (<= y 4.6e-11) (* 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 <= -2.06e-10) {
		tmp = t_0;
	} else if (y <= 4.6e-11) {
		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 <= (-2.06d-10)) then
        tmp = t_0
    else if (y <= 4.6d-11) 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 <= -2.06e-10) {
		tmp = t_0;
	} else if (y <= 4.6e-11) {
		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 <= -2.06e-10:
		tmp = t_0
	elif y <= 4.6e-11:
		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 <= -2.06e-10)
		tmp = t_0;
	elseif (y <= 4.6e-11)
		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 <= -2.06e-10)
		tmp = t_0;
	elseif (y <= 4.6e-11)
		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, -2.06e-10], t$95$0, If[LessEqual[y, 4.6e-11], N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0600000000000001e-10 or 4.60000000000000027e-11 < y

   1. Initial program 97.7%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot \left(z - x\right)\right)}\right) \]

      7. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(-1 \cdot z - \color{blue}{-1 \cdot x}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot x\right)\right)}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(-1 \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(-1 \cdot z + x\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{-1 \cdot z}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(z\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(x - \color{blue}{z}\right)\right) \]

      14. --lowering--.f64 97.7%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(x, \color{blue}{z}\right)\right) \]

   5. Simplified 97.7%

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

   if -2.0600000000000001e-10 < y < 4.60000000000000027e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(1 - y\right)}\right) \]

      2. --lowering--.f64 73.1%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(1, \color{blue}{y}\right)\right) \]

   5. Simplified 73.1%

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

Alternative 4: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x - z\right)\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- x z))))
   (if (<= y -4.3e-12) t_0 (if (<= y 2.6e-17) z t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x - z);
	double tmp;
	if (y <= -4.3e-12) {
		tmp = t_0;
	} else if (y <= 2.6e-17) {
		tmp = z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x - z)
    if (y <= (-4.3d-12)) then
        tmp = t_0
    else if (y <= 2.6d-17) then
        tmp = z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (x - z);
	double tmp;
	if (y <= -4.3e-12) {
		tmp = t_0;
	} else if (y <= 2.6e-17) {
		tmp = z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x - z)
	tmp = 0
	if y <= -4.3e-12:
		tmp = t_0
	elif y <= 2.6e-17:
		tmp = z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x - z))
	tmp = 0.0
	if (y <= -4.3e-12)
		tmp = t_0;
	elseif (y <= 2.6e-17)
		tmp = z;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x - z);
	tmp = 0.0;
	if (y <= -4.3e-12)
		tmp = t_0;
	elseif (y <= 2.6e-17)
		tmp = z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.3e-12], t$95$0, If[LessEqual[y, 2.6e-17], z, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.29999999999999985e-12 or 2.60000000000000003e-17 < y

   1. Initial program 97.7%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot \left(z - x\right)\right)}\right) \]

      7. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(-1 \cdot z - \color{blue}{-1 \cdot x}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot x\right)\right)}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(-1 \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(-1 \cdot z + x\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{-1 \cdot z}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(z\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(x - \color{blue}{z}\right)\right) \]

      14. --lowering--.f64 97.0%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(x, \color{blue}{z}\right)\right) \]

   5. Simplified 97.0%

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

   if -4.29999999999999985e-12 < y < 2.60000000000000003e-17

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Simplified 73.3%

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

Alternative 5: 60.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-12}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.1e-12) (* x y) (if (<= y 2.1e-16) z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.1e-12) {
		tmp = x * y;
	} else if (y <= 2.1e-16) {
		tmp = z;
	} else {
		tmp = x * 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 <= (-1.1d-12)) then
        tmp = x * y
    else if (y <= 2.1d-16) then
        tmp = z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.1e-12) {
		tmp = x * y;
	} else if (y <= 2.1e-16) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.1e-12:
		tmp = x * y
	elif y <= 2.1e-16:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.1e-12)
		tmp = Float64(x * y);
	elseif (y <= 2.1e-16)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.1e-12)
		tmp = x * y;
	elseif (y <= 2.1e-16)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.1e-12], N[(x * y), $MachinePrecision], If[LessEqual[y, 2.1e-16], z, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.09999999999999996e-12 or 2.1000000000000001e-16 < y

   1. Initial program 97.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 50.7%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 50.7%

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

   if -1.09999999999999996e-12 < y < 2.1000000000000001e-16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Simplified 73.3%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-12}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. +-commutative N/A

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

   2. distribute-rgt-out-- N/A

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

   3. *-lft-identity N/A

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

   4. associate-+l- N/A

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

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \color{blue}{\left(y \cdot z - x \cdot y\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \left(y \cdot z - y \cdot \color{blue}{x}\right)\right) \]

   7. distribute-lft-out-- N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \left(y \cdot \color{blue}{\left(z - x\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(y, \color{blue}{\left(z - x\right)}\right)\right) \]

   9. --lowering--.f64 100.0%

      \[\leadsto \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(z, \color{blue}{x}\right)\right)\right) \]

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 7: 36.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{z} \]
4. Step-by-step derivation

5. Simplified 37.4%

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

Developer Target 1: 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 2024150 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment:bezierClip from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- z (* (- z x) y)))

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