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

Percentage Accurate: 88.4% → 99.7%

Time: 9.3s

Alternatives: 10

Speedup: 0.7×

• 21.0% 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: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-34} \lor \neg \left(z \leq 2 \cdot 10^{-44}\right):\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1e-34) (not (<= z 2e-44)))
   (* x (/ (+ (- y z) 1.0) z))
   (/ (* x (+ y 1.0)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-34) || !(z <= 2e-44)) {
		tmp = x * (((y - z) + 1.0) / z);
	} else {
		tmp = (x * (y + 1.0)) / 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 ((z <= (-1d-34)) .or. (.not. (z <= 2d-44))) then
        tmp = x * (((y - z) + 1.0d0) / z)
    else
        tmp = (x * (y + 1.0d0)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-34) || !(z <= 2e-44)) {
		tmp = x * (((y - z) + 1.0) / z);
	} else {
		tmp = (x * (y + 1.0)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1e-34) or not (z <= 2e-44):
		tmp = x * (((y - z) + 1.0) / z)
	else:
		tmp = (x * (y + 1.0)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1e-34) || !(z <= 2e-44))
		tmp = Float64(x * Float64(Float64(Float64(y - z) + 1.0) / z));
	else
		tmp = Float64(Float64(x * Float64(y + 1.0)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1e-34) || ~((z <= 2e-44)))
		tmp = x * (((y - z) + 1.0) / z);
	else
		tmp = (x * (y + 1.0)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1e-34], N[Not[LessEqual[z, 2e-44]], $MachinePrecision]], N[(x * N[(N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999928e-35 or 1.99999999999999991e-44 < z

   1. Initial program 79.1%

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

   2. Taylor expanded in x around 0 79.1%

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

      1. associate--l+ 79.1%

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

      2. +-commutative 79.1%

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

      3. associate-*r/ 99.9%

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

      4. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   if -9.99999999999999928e-35 < z < 1.99999999999999991e-44

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-34} \lor \neg \left(z \leq 2 \cdot 10^{-44}\right):\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \end{array} \]

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;\frac{x \cdot t_0}{z} \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- y z) 1.0)))
   (if (<= (/ (* x t_0) z) 2e+158) (/ x (/ z t_0)) (/ t_0 (/ z x)))))

C

double code(double x, double y, double z) {
	double t_0 = (y - z) + 1.0;
	double tmp;
	if (((x * t_0) / z) <= 2e+158) {
		tmp = x / (z / t_0);
	} else {
		tmp = t_0 / (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) :: t_0
    real(8) :: tmp
    t_0 = (y - z) + 1.0d0
    if (((x * t_0) / z) <= 2d+158) then
        tmp = x / (z / t_0)
    else
        tmp = t_0 / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - z) + 1.0;
	double tmp;
	if (((x * t_0) / z) <= 2e+158) {
		tmp = x / (z / t_0);
	} else {
		tmp = t_0 / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - z) + 1.0
	tmp = 0
	if ((x * t_0) / z) <= 2e+158:
		tmp = x / (z / t_0)
	else:
		tmp = t_0 / (z / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - z) + 1.0)
	tmp = 0.0
	if (Float64(Float64(x * t_0) / z) <= 2e+158)
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(t_0 / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - z) + 1.0;
	tmp = 0.0;
	if (((x * t_0) / z) <= 2e+158)
		tmp = x / (z / t_0);
	else
		tmp = t_0 / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision], 2e+158], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(z / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 1.99999999999999991e158

   1. Initial program 88.8%

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

   2. Taylor expanded in x around 0 88.8%

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

      1. associate--l+ 88.8%

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

      2. +-commutative 88.8%

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

      3. associate-*r/ 97.9%

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

      4. +-commutative 97.9%

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

   4. Simplified 97.9%

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

      1. clear-num 97.8%

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

      2. un-div-inv 97.9%

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

   6. Applied egg-rr 97.9%

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

   if 1.99999999999999991e158 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

   1. Initial program 84.7%

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

   2. Taylor expanded in x around 0 84.7%

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

      1. associate--l+ 84.7%

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

      2. +-commutative 84.7%

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

      3. associate-*r/ 90.7%

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

      4. +-commutative 90.7%

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

   4. Simplified 90.7%

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

      1. associate-*r/ 84.7%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

      4. clear-num 99.8%

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

      5. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - z\right) + 1}{\frac{z}{x}}\\ \end{array} \]

Alternative 3: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;\frac{x \cdot t_0}{z} \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- y z) 1.0)))
   (if (<= (/ (* x t_0) z) 5e-89) (/ x (/ z t_0)) (* t_0 (/ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = (y - z) + 1.0;
	double tmp;
	if (((x * t_0) / z) <= 5e-89) {
		tmp = x / (z / t_0);
	} else {
		tmp = t_0 * (x / 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) :: t_0
    real(8) :: tmp
    t_0 = (y - z) + 1.0d0
    if (((x * t_0) / z) <= 5d-89) then
        tmp = x / (z / t_0)
    else
        tmp = t_0 * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - z) + 1.0;
	double tmp;
	if (((x * t_0) / z) <= 5e-89) {
		tmp = x / (z / t_0);
	} else {
		tmp = t_0 * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - z) + 1.0
	tmp = 0
	if ((x * t_0) / z) <= 5e-89:
		tmp = x / (z / t_0)
	else:
		tmp = t_0 * (x / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - z) + 1.0)
	tmp = 0.0
	if (Float64(Float64(x * t_0) / z) <= 5e-89)
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(t_0 * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - z) + 1.0;
	tmp = 0.0;
	if (((x * t_0) / z) <= 5e-89)
		tmp = x / (z / t_0);
	else
		tmp = t_0 * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision], 5e-89], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 4.99999999999999967e-89

   1. Initial program 87.1%

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

   2. Taylor expanded in x around 0 87.1%

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

      1. associate--l+ 87.1%

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

      2. +-commutative 87.1%

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

      3. associate-*r/ 97.6%

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

      4. +-commutative 97.6%

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

   4. Simplified 97.6%

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

      1. clear-num 97.5%

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

      2. un-div-inv 97.7%

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

   6. Applied egg-rr 97.7%

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

   if 4.99999999999999967e-89 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

   1. Initial program 89.3%

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

      1. associate-/l* 93.5%

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

      2. associate-/r/ 99.8%

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

   3. Applied egg-rr 99.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4: 63.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -4.7 \cdot 10^{+126}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-90}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -8.3 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-171}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))))
   (if (<= z -4.7e+126)
     (- x)
     (if (<= z -1.7e-90)
       (* x (/ y z))
       (if (<= z -8.3e-137)
         (/ x z)
         (if (<= z -1.15e-171)
           t_0
           (if (<= z 6.8e-71) (/ x z) (if (<= z 4.5e+46) t_0 (- x)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (z <= -4.7e+126) {
		tmp = -x;
	} else if (z <= -1.7e-90) {
		tmp = x * (y / z);
	} else if (z <= -8.3e-137) {
		tmp = x / z;
	} else if (z <= -1.15e-171) {
		tmp = t_0;
	} else if (z <= 6.8e-71) {
		tmp = x / z;
	} else if (z <= 4.5e+46) {
		tmp = t_0;
	} else {
		tmp = -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 * (x / z)
    if (z <= (-4.7d+126)) then
        tmp = -x
    else if (z <= (-1.7d-90)) then
        tmp = x * (y / z)
    else if (z <= (-8.3d-137)) then
        tmp = x / z
    else if (z <= (-1.15d-171)) then
        tmp = t_0
    else if (z <= 6.8d-71) then
        tmp = x / z
    else if (z <= 4.5d+46) then
        tmp = t_0
    else
        tmp = -x
    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 (z <= -4.7e+126) {
		tmp = -x;
	} else if (z <= -1.7e-90) {
		tmp = x * (y / z);
	} else if (z <= -8.3e-137) {
		tmp = x / z;
	} else if (z <= -1.15e-171) {
		tmp = t_0;
	} else if (z <= 6.8e-71) {
		tmp = x / z;
	} else if (z <= 4.5e+46) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / z)
	tmp = 0
	if z <= -4.7e+126:
		tmp = -x
	elif z <= -1.7e-90:
		tmp = x * (y / z)
	elif z <= -8.3e-137:
		tmp = x / z
	elif z <= -1.15e-171:
		tmp = t_0
	elif z <= 6.8e-71:
		tmp = x / z
	elif z <= 4.5e+46:
		tmp = t_0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (z <= -4.7e+126)
		tmp = Float64(-x);
	elseif (z <= -1.7e-90)
		tmp = Float64(x * Float64(y / z));
	elseif (z <= -8.3e-137)
		tmp = Float64(x / z);
	elseif (z <= -1.15e-171)
		tmp = t_0;
	elseif (z <= 6.8e-71)
		tmp = Float64(x / z);
	elseif (z <= 4.5e+46)
		tmp = t_0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x / z);
	tmp = 0.0;
	if (z <= -4.7e+126)
		tmp = -x;
	elseif (z <= -1.7e-90)
		tmp = x * (y / z);
	elseif (z <= -8.3e-137)
		tmp = x / z;
	elseif (z <= -1.15e-171)
		tmp = t_0;
	elseif (z <= 6.8e-71)
		tmp = x / z;
	elseif (z <= 4.5e+46)
		tmp = t_0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.7e+126], (-x), If[LessEqual[z, -1.7e-90], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -8.3e-137], N[(x / z), $MachinePrecision], If[LessEqual[z, -1.15e-171], t$95$0, If[LessEqual[z, 6.8e-71], N[(x / z), $MachinePrecision], If[LessEqual[z, 4.5e+46], t$95$0, (-x)]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.6999999999999999e126 or 4.5000000000000001e46 < z

   1. Initial program 73.3%

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

   2. Taylor expanded in z around inf 83.4%

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

      1. neg-mul-1 83.4%

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

   4. Simplified 83.4%

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

   if -4.6999999999999999e126 < z < -1.69999999999999997e-90

   1. Initial program 88.3%

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

   2. Taylor expanded in x around 0 88.3%

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

      1. associate--l+ 88.3%

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

      2. +-commutative 88.3%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in y around inf 50.2%

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

      1. associate-*r/ 55.9%

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

   7. Simplified 55.9%

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

   if -1.69999999999999997e-90 < z < -8.30000000000000018e-137 or -1.14999999999999989e-171 < z < 6.80000000000000007e-71

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 72.5%

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

      1. associate-/l* 72.5%

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

      2. associate-/r/ 72.5%

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

   4. Simplified 72.5%

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

   5. Taylor expanded in z around 0 72.5%

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

   if -8.30000000000000018e-137 < z < -1.14999999999999989e-171 or 6.80000000000000007e-71 < z < 4.5000000000000001e46

   1. Initial program 97.3%

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

   2. Taylor expanded in y around inf 70.2%

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

      1. associate-/l* 65.1%

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

      2. associate-/r/ 75.0%

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

   4. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+126}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-90}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -8.3 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-171}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 5: 63.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -4.7 \cdot 10^{+126}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ y z))))
   (if (<= z -4.7e+126)
     (- x)
     (if (<= z -2e-87)
       t_0
       (if (<= z 1.02e-70) (/ x z) (if (<= z 5.8e+47) t_0 (- x)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y / z);
	double tmp;
	if (z <= -4.7e+126) {
		tmp = -x;
	} else if (z <= -2e-87) {
		tmp = t_0;
	} else if (z <= 1.02e-70) {
		tmp = x / z;
	} else if (z <= 5.8e+47) {
		tmp = t_0;
	} else {
		tmp = -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 = x * (y / z)
    if (z <= (-4.7d+126)) then
        tmp = -x
    else if (z <= (-2d-87)) then
        tmp = t_0
    else if (z <= 1.02d-70) then
        tmp = x / z
    else if (z <= 5.8d+47) then
        tmp = t_0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y / z);
	double tmp;
	if (z <= -4.7e+126) {
		tmp = -x;
	} else if (z <= -2e-87) {
		tmp = t_0;
	} else if (z <= 1.02e-70) {
		tmp = x / z;
	} else if (z <= 5.8e+47) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y / z)
	tmp = 0
	if z <= -4.7e+126:
		tmp = -x
	elif z <= -2e-87:
		tmp = t_0
	elif z <= 1.02e-70:
		tmp = x / z
	elif z <= 5.8e+47:
		tmp = t_0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y / z))
	tmp = 0.0
	if (z <= -4.7e+126)
		tmp = Float64(-x);
	elseif (z <= -2e-87)
		tmp = t_0;
	elseif (z <= 1.02e-70)
		tmp = Float64(x / z);
	elseif (z <= 5.8e+47)
		tmp = t_0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y / z);
	tmp = 0.0;
	if (z <= -4.7e+126)
		tmp = -x;
	elseif (z <= -2e-87)
		tmp = t_0;
	elseif (z <= 1.02e-70)
		tmp = x / z;
	elseif (z <= 5.8e+47)
		tmp = t_0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.7e+126], (-x), If[LessEqual[z, -2e-87], t$95$0, If[LessEqual[z, 1.02e-70], N[(x / z), $MachinePrecision], If[LessEqual[z, 5.8e+47], t$95$0, (-x)]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.6999999999999999e126 or 5.79999999999999961e47 < z

   1. Initial program 73.3%

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

   2. Taylor expanded in z around inf 83.4%

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

      1. neg-mul-1 83.4%

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

   4. Simplified 83.4%

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

   if -4.6999999999999999e126 < z < -2.00000000000000004e-87 or 1.0200000000000001e-70 < z < 5.79999999999999961e47

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0 91.6%

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

      1. associate--l+ 91.6%

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

      2. +-commutative 91.6%

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

      3. associate-*r/ 97.4%

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

      4. +-commutative 97.4%

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

   4. Simplified 97.4%

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

   5. Taylor expanded in y around inf 56.5%

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

      1. associate-*r/ 58.8%

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

   7. Simplified 58.8%

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

   if -2.00000000000000004e-87 < z < 1.0200000000000001e-70

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 69.1%

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

      1. associate-/l* 69.1%

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

      2. associate-/r/ 69.1%

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

   4. Simplified 69.1%

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

   5. Taylor expanded in z around 0 69.1%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+126}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-87}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+47}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 6: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -40000000000 \lor \neg \left(y \leq 7.8 \cdot 10^{+44}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -40000000000.0) (not (<= y 7.8e+44)))
   (* y (/ x z))
   (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -40000000000.0) || !(y <= 7.8e+44)) {
		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 <= (-40000000000.0d0)) .or. (.not. (y <= 7.8d+44))) 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 <= -40000000000.0) || !(y <= 7.8e+44)) {
		tmp = y * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -40000000000.0) or not (y <= 7.8e+44):
		tmp = y * (x / z)
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -40000000000.0) || !(y <= 7.8e+44))
		tmp = Float64(y * Float64(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 <= -40000000000.0) || ~((y <= 7.8e+44)))
		tmp = y * (x / z);
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -40000000000.0], N[Not[LessEqual[y, 7.8e+44]], $MachinePrecision]], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4e10 or 7.8000000000000005e44 < y

   1. Initial program 89.5%

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

   2. Taylor expanded in y around inf 75.5%

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

      1. associate-/l* 72.7%

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

      2. associate-/r/ 75.7%

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

   4. Simplified 75.7%

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

   if -4e10 < y < 7.8000000000000005e44

   1. Initial program 86.6%

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

   2. Taylor expanded in x around 0 86.6%

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

      1. associate--l+ 86.6%

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

      2. +-commutative 86.6%

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

      3. associate-*r/ 99.9%

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

      4. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around 0 85.0%

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

      1. *-commutative 85.0%

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

      2. associate-/l* 83.1%

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

      3. div-sub 78.4%

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

      4. associate-/r/ 78.4%

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

      5. associate-*l/ 78.6%

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

      6. *-lft-identity 78.6%

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

      7. associate-/r/ 98.4%

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

      8. *-inverses 98.4%

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

      9. *-lft-identity 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -40000000000 \lor \neg \left(y \leq 7.8 \cdot 10^{+44}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 7: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2000000000000:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2000000000000.0)
   (/ (* x y) z)
   (if (<= y 4.8e+43) (- (/ x z) x) (* y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2000000000000.0) {
		tmp = (x * y) / z;
	} else if (y <= 4.8e+43) {
		tmp = (x / z) - x;
	} else {
		tmp = y * (x / 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 <= (-2000000000000.0d0)) then
        tmp = (x * y) / z
    else if (y <= 4.8d+43) then
        tmp = (x / z) - x
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2000000000000.0) {
		tmp = (x * y) / z;
	} else if (y <= 4.8e+43) {
		tmp = (x / z) - x;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2000000000000.0:
		tmp = (x * y) / z
	elif y <= 4.8e+43:
		tmp = (x / z) - x
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2000000000000.0)
		tmp = Float64(Float64(x * y) / z);
	elseif (y <= 4.8e+43)
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2000000000000.0)
		tmp = (x * y) / z;
	elseif (y <= 4.8e+43)
		tmp = (x / z) - x;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2000000000000.0], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 4.8e+43], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e12

   1. Initial program 92.7%

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

   2. Taylor expanded in y around inf 73.9%

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

   if -2e12 < y < 4.80000000000000046e43

   1. Initial program 86.6%

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

   2. Taylor expanded in x around 0 86.6%

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

      1. associate--l+ 86.6%

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

      2. +-commutative 86.6%

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

      3. associate-*r/ 99.9%

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

      4. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around 0 85.0%

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

      1. *-commutative 85.0%

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

      2. associate-/l* 83.1%

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

      3. div-sub 78.4%

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

      4. associate-/r/ 78.4%

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

      5. associate-*l/ 78.6%

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

      6. *-lft-identity 78.6%

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

      7. associate-/r/ 98.4%

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

      8. *-inverses 98.4%

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

      9. *-lft-identity 98.4%

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

   7. Simplified 98.4%

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

   if 4.80000000000000046e43 < y

   1. Initial program 86.3%

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

   2. Taylor expanded in y around inf 76.9%

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

      1. associate-/l* 76.7%

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

      2. associate-/r/ 79.7%

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

   4. Simplified 79.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2000000000000:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 8: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -51000000000000:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -51000000000000.0)
   (/ (* x (+ y 1.0)) z)
   (if (<= y 4.4e+43) (- (/ x z) x) (* y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -51000000000000.0) {
		tmp = (x * (y + 1.0)) / z;
	} else if (y <= 4.4e+43) {
		tmp = (x / z) - x;
	} else {
		tmp = y * (x / 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 <= (-51000000000000.0d0)) then
        tmp = (x * (y + 1.0d0)) / z
    else if (y <= 4.4d+43) then
        tmp = (x / z) - x
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -51000000000000.0) {
		tmp = (x * (y + 1.0)) / z;
	} else if (y <= 4.4e+43) {
		tmp = (x / z) - x;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -51000000000000.0:
		tmp = (x * (y + 1.0)) / z
	elif y <= 4.4e+43:
		tmp = (x / z) - x
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -51000000000000.0)
		tmp = Float64(Float64(x * Float64(y + 1.0)) / z);
	elseif (y <= 4.4e+43)
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -51000000000000.0)
		tmp = (x * (y + 1.0)) / z;
	elseif (y <= 4.4e+43)
		tmp = (x / z) - x;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -51000000000000.0], N[(N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 4.4e+43], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.1e13

   1. Initial program 92.7%

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

   2. Taylor expanded in z around 0 74.0%

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

   if -5.1e13 < y < 4.40000000000000001e43

   1. Initial program 86.6%

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

   2. Taylor expanded in x around 0 86.6%

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

      1. associate--l+ 86.6%

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

      2. +-commutative 86.6%

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

      3. associate-*r/ 99.9%

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

      4. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around 0 85.0%

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

      1. *-commutative 85.0%

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

      2. associate-/l* 83.1%

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

      3. div-sub 78.4%

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

      4. associate-/r/ 78.4%

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

      5. associate-*l/ 78.6%

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

      6. *-lft-identity 78.6%

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

      7. associate-/r/ 98.4%

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

      8. *-inverses 98.4%

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

      9. *-lft-identity 98.4%

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

   7. Simplified 98.4%

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

   if 4.40000000000000001e43 < y

   1. Initial program 86.3%

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

   2. Taylor expanded in y around inf 76.9%

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

      1. associate-/l* 76.7%

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

      2. associate-/r/ 79.7%

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

   4. Simplified 79.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -51000000000000:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 9: 65.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 980000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0) (- x) (if (<= z 980000.0) (/ x z) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = -x;
	} else if (z <= 980000.0) {
		tmp = x / z;
	} else {
		tmp = -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 (z <= (-1.0d0)) then
        tmp = -x
    else if (z <= 980000.0d0) then
        tmp = x / z
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = -x;
	} else if (z <= 980000.0) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = -x
	elif z <= 980000.0:
		tmp = x / z
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(-x);
	elseif (z <= 980000.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = -x;
	elseif (z <= 980000.0)
		tmp = x / z;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], (-x), If[LessEqual[z, 980000.0], N[(x / z), $MachinePrecision], (-x)]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 9.8e5 < z

   1. Initial program 75.1%

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

   2. Taylor expanded in z around inf 72.7%

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

      1. neg-mul-1 72.7%

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

   4. Simplified 72.7%

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

   if -1 < z < 9.8e5

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 59.6%

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

      1. associate-/l* 59.6%

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

      2. associate-/r/ 59.7%

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

   4. Simplified 59.7%

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

   5. Taylor expanded in z around 0 56.9%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 980000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 10: 39.0% accurate, 4.5× 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 87.8%

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

2. Taylor expanded in z around inf 37.3%

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

   1. neg-mul-1 37.3%

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

4. Simplified 37.3%

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

5. Final simplification 37.3%

   \[\leadsto -x \]

Developer target: 99.6% 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 2023275 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

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