Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Percentage Accurate: 88.4% → 96.1%

Time: 6.0s

Alternatives: 6

Speedup: 1.1×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / 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 = (x * ((y - z) + 1.0d0)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / 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 = (x * ((y - z) + 1.0d0)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.0%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 90.0

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

4. Applied rewrites 90.0%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. distribute-lft1-in N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lift-+.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift--.f64 N/A

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

   10. div-add N/A

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

   11. frac-add N/A

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

   12. associate-*l/ N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-lft-identity N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 57.6

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

6. Applied rewrites 57.6%

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

7. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-/r* N/A

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

   3. distribute-lft-in N/A

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

   4. div-add N/A

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

   5. associate-/l* N/A

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

   6. *-inverses N/A

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

   7. *-rgt-identity N/A

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

   8. div-add-rev N/A

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

   9. *-rgt-identity N/A

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

   10. associate-*r/ N/A

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

   11. associate-/r* N/A

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

   12. associate-*r* N/A

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

   13. times-frac N/A

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

   14. associate-/l* N/A

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

   15. *-inverses N/A

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

   16. *-rgt-identity N/A

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

   17. distribute-lft-in N/A

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

   18. div-add N/A

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

   19. associate--l+ N/A

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

9. Applied rewrites 98.4%

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

10. Final simplification 98.4%

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

Alternative 2: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+49} \lor \neg \left(z \leq 4.8 \cdot 10^{+84}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.2e+49) (not (<= z 4.8e+84))) (- x) (/ (fma y x x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2e+49) || !(z <= 4.8e+84)) {
		tmp = -x;
	} else {
		tmp = fma(y, x, x) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.2e+49) || !(z <= 4.8e+84))
		tmp = Float64(-x);
	else
		tmp = Float64(fma(y, x, x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.2e+49], N[Not[LessEqual[z, 4.8e+84]], $MachinePrecision]], (-x), N[(N[(y * x + x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.19999999999999977e49 or 4.7999999999999999e84 < z

   1. Initial program 74.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 84.8

         \[\leadsto \color{blue}{-x} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{-x} \]

   if -5.19999999999999977e49 < z < 4.7999999999999999e84

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 94.1

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

   5. Applied rewrites 94.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, x\right)}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+49} \lor \neg \left(z \leq 4.8 \cdot 10^{+84}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+116} \lor \neg \left(y \leq 10^{+35}\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.5e+116) (not (<= y 1e+35))) (/ (* y x) z) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.5e+116) || !(y <= 1e+35)) {
		tmp = (y * x) / z;
	} else {
		tmp = (x / z) - 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 <= (-3.5d+116)) .or. (.not. (y <= 1d+35))) then
        tmp = (y * x) / z
    else
        tmp = (x / z) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.5e+116) || !(y <= 1e+35)) {
		tmp = (y * x) / z;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.5e+116) or not (y <= 1e+35):
		tmp = (y * x) / z
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.5e+116) || !(y <= 1e+35))
		tmp = Float64(Float64(y * x) / z);
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.5e+116) || ~((y <= 1e+35)))
		tmp = (y * x) / z;
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.5e+116], N[Not[LessEqual[y, 1e+35]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.49999999999999997e116 or 9.9999999999999997e34 < y

   1. Initial program 95.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -3.49999999999999997e116 < y < 9.9999999999999997e34

   1. Initial program 86.5%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{1 - z}{z}} \]

      2. div-sub N/A

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

      3. *-inverses N/A

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

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

         \[\leadsto \color{blue}{x \cdot \frac{1}{z} - x \cdot 1} \]

      5. *-rgt-identity N/A

         \[\leadsto x \cdot \frac{1}{z} - \color{blue}{x} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot 1}{z}} - x \]

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 90.2

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

   5. Applied rewrites 90.2%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+116} \lor \neg \left(y \leq 10^{+35}\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+116} \lor \neg \left(y \leq 10^{+35}\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.5e+116) (not (<= y 1e+35))) (* (/ x z) y) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.5e+116) || !(y <= 1e+35)) {
		tmp = (x / z) * y;
	} else {
		tmp = (x / z) - 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 <= (-3.5d+116)) .or. (.not. (y <= 1d+35))) then
        tmp = (x / z) * y
    else
        tmp = (x / z) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.5e+116) || !(y <= 1e+35)) {
		tmp = (x / z) * y;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.5e+116) or not (y <= 1e+35):
		tmp = (x / z) * y
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.5e+116) || !(y <= 1e+35))
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.5e+116) || ~((y <= 1e+35)))
		tmp = (x / z) * y;
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.5e+116], N[Not[LessEqual[y, 1e+35]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.49999999999999997e116 or 9.9999999999999997e34 < y

   1. Initial program 95.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 81.8

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

   7. Applied rewrites 81.8%

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

   if -3.49999999999999997e116 < y < 9.9999999999999997e34

   1. Initial program 86.5%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{1 - z}{z}} \]

      2. div-sub N/A

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

      3. *-inverses N/A

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

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

         \[\leadsto \color{blue}{x \cdot \frac{1}{z} - x \cdot 1} \]

      5. *-rgt-identity N/A

         \[\leadsto x \cdot \frac{1}{z} - \color{blue}{x} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot 1}{z}} - x \]

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 90.2

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

   5. Applied rewrites 90.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+116} \lor \neg \left(y \leq 10^{+35}\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{x}{z} - x \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.0%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{1 - z}{z}} \]

   2. div-sub N/A

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

   3. *-inverses N/A

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

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

      \[\leadsto \color{blue}{x \cdot \frac{1}{z} - x \cdot 1} \]

   5. *-rgt-identity N/A

      \[\leadsto x \cdot \frac{1}{z} - \color{blue}{x} \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{x \cdot 1}{z}} - x \]

   7. *-rgt-identity N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 63.7

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

5. Applied rewrites 63.7%

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

Alternative 6: 38.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-x)

Derivation

1. Initial program 90.0%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{-1 \cdot x} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f64 38.1

      \[\leadsto \color{blue}{-x} \]

5. Applied rewrites 38.1%

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

Developer Target 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (+ 1.0 y) (/ x z)) x)))
   (if (< x -2.71483106713436e-162)
     t_0
     (if (< x 3.874108816439546e-197)
       (* (* x (+ (- y z) 1.0)) (/ 1.0 z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / 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 = ((1.0d0 + y) * (x / z)) - x
    if (x < (-2.71483106713436d-162)) then
        tmp = t_0
    else if (x < 3.874108816439546d-197) then
        tmp = (x * ((y - z) + 1.0d0)) * (1.0d0 / 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 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((1.0 + y) * (x / z)) - x
	tmp = 0
	if x < -2.71483106713436e-162:
		tmp = t_0
	elif x < 3.874108816439546e-197:
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 + y) * Float64(x / z)) - x)
	tmp = 0.0
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = Float64(Float64(x * Float64(Float64(y - z) + 1.0)) * Float64(1.0 / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((1.0 + y) * (x / z)) - x;
	tmp = 0.0;
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 + y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[Less[x, -2.71483106713436e-162], t$95$0, If[Less[x, 3.874108816439546e-197], N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024327 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -67870776678359/25000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (+ 1 y) (/ x z)) x) (if (< x 1937054408219773/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x))))

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