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

Percentage Accurate: 88.5% → 99.6%

Time: 6.9s

Alternatives: 13

Speedup: 0.8×

• 20.4% 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: 88.5% 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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+57} \lor \neg \left(z \leq 8.5\right):\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9.5e+57) (not (<= z 8.5)))
   (- (* x (/ y z)) x)
   (* (/ x z) (- (+ y 1.0) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.5e+57) || !(z <= 8.5)) {
		tmp = (x * (y / z)) - x;
	} else {
		tmp = (x / z) * ((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 <= (-9.5d+57)) .or. (.not. (z <= 8.5d0))) then
        tmp = (x * (y / z)) - x
    else
        tmp = (x / z) * ((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 <= -9.5e+57) || !(z <= 8.5)) {
		tmp = (x * (y / z)) - x;
	} else {
		tmp = (x / z) * ((y + 1.0) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9.5e+57) or not (z <= 8.5):
		tmp = (x * (y / z)) - x
	else:
		tmp = (x / z) * ((y + 1.0) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9.5e+57) || !(z <= 8.5))
		tmp = Float64(Float64(x * Float64(y / z)) - x);
	else
		tmp = Float64(Float64(x / z) * Float64(Float64(y + 1.0) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -9.5e+57) || ~((z <= 8.5)))
		tmp = (x * (y / z)) - x;
	else
		tmp = (x / z) * ((y + 1.0) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9.5e+57], N[Not[LessEqual[z, 8.5]], $MachinePrecision]], N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4999999999999997e57 or 8.5 < z

   1. Initial program 81.4%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in y around inf 92.4%

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

      1. associate-/l* 94.7%

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

      2. associate-/r/ 99.9%

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

   6. Simplified 99.9%

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

   if -9.4999999999999997e57 < z < 8.5

   1. Initial program 98.4%

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

      1. associate-/l* 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in x around 0 98.4%

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

      1. associate-*l/ 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+57} \lor \neg \left(z \leq 8.5\right):\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \]

Alternative 2: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-55}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1e-55) (- (/ (fma x y x) z) x) (/ x (/ z (+ (- y z) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e-55) {
		tmp = (fma(x, y, x) / z) - x;
	} else {
		tmp = x / (z / ((y - z) + 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1e-55)
		tmp = Float64(Float64(fma(x, y, x) / z) - x);
	else
		tmp = Float64(x / Float64(z / Float64(Float64(y - z) + 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1e-55], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], N[(x / N[(z / N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999995e-56

   1. Initial program 94.4%

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

   3. Simplified 99.3%

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

   if 9.99999999999999995e-56 < x

   1. Initial program 80.3%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-55}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array} \]

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;\frac{x \cdot t_0}{z} \leq 10^{-23}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- y z) 1.0)))
   (if (<= (/ (* x t_0) z) 1e-23)
     (/ x (/ z t_0))
     (* (/ x z) (- (+ y 1.0) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (y - z) + 1.0;
	double tmp;
	if (((x * t_0) / z) <= 1e-23) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x / z) * ((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) :: t_0
    real(8) :: tmp
    t_0 = (y - z) + 1.0d0
    if (((x * t_0) / z) <= 1d-23) then
        tmp = x / (z / t_0)
    else
        tmp = (x / z) * ((y + 1.0d0) - 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) <= 1e-23) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x / z) * ((y + 1.0) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - z) + 1.0
	tmp = 0
	if ((x * t_0) / z) <= 1e-23:
		tmp = x / (z / t_0)
	else:
		tmp = (x / z) * ((y + 1.0) - 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) <= 1e-23)
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(Float64(x / z) * Float64(Float64(y + 1.0) - 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) <= 1e-23)
		tmp = x / (z / t_0);
	else
		tmp = (x / z) * ((y + 1.0) - 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], 1e-23], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 9.9999999999999996e-24

   1. Initial program 91.4%

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

      1. associate-/l* 97.6%

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

   3. Simplified 97.6%

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

   if 9.9999999999999996e-24 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

   1. Initial program 87.2%

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

      1. associate-/l* 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in x around 0 87.2%

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

      1. associate-*l/ 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 10^{-23}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y + 1\right) - z\right)\\ \end{array} \]

Alternative 4: 65.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-219}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 2.22 \cdot 10^{-94}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 750000000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (- x)
   (if (<= z -2e-219)
     (/ x z)
     (if (<= z 2.22e-94)
       (* y (/ x z))
       (if (<= z 750000000000.0) (/ x z) (- x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = -x;
	} else if (z <= -2e-219) {
		tmp = x / z;
	} else if (z <= 2.22e-94) {
		tmp = y * (x / z);
	} else if (z <= 750000000000.0) {
		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 <= (-2d-219)) then
        tmp = x / z
    else if (z <= 2.22d-94) then
        tmp = y * (x / z)
    else if (z <= 750000000000.0d0) 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 <= -2e-219) {
		tmp = x / z;
	} else if (z <= 2.22e-94) {
		tmp = y * (x / z);
	} else if (z <= 750000000000.0) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = -x
	elif z <= -2e-219:
		tmp = x / z
	elif z <= 2.22e-94:
		tmp = y * (x / z)
	elif z <= 750000000000.0:
		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 <= -2e-219)
		tmp = Float64(x / z);
	elseif (z <= 2.22e-94)
		tmp = Float64(y * Float64(x / z));
	elseif (z <= 750000000000.0)
		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 <= -2e-219)
		tmp = x / z;
	elseif (z <= 2.22e-94)
		tmp = y * (x / z);
	elseif (z <= 750000000000.0)
		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, -2e-219], N[(x / z), $MachinePrecision], If[LessEqual[z, 2.22e-94], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 750000000000.0], N[(x / z), $MachinePrecision], (-x)]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 7.5e11 < z

   1. Initial program 81.5%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in z around inf 75.0%

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

      1. neg-mul-1 75.0%

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

   6. Simplified 75.0%

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

   if -1 < z < -2.0000000000000001e-219 or 2.2200000000000001e-94 < z < 7.5e11

   1. Initial program 99.8%

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

      1. associate-/l* 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in y around 0 64.5%

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

      1. associate-/l* 64.3%

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

      2. associate-/r/ 64.4%

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

   6. Simplified 64.4%

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

   7. Taylor expanded in z around 0 62.6%

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

   if -2.0000000000000001e-219 < z < 2.2200000000000001e-94

   1. Initial program 99.8%

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

      1. associate-/l* 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in y around inf 55.1%

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

      1. associate-/r/ 71.9%

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

   6. Applied egg-rr 71.9%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-219}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 2.22 \cdot 10^{-94}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 750000000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 5: 95.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (- (* x (/ y z)) x) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = (x * (y / z)) - x;
	} 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 <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = (x * (y / z)) - x
    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 <= -1.0) || !(y <= 1.0)) {
		tmp = (x * (y / z)) - x;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 87.1%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in y around inf 91.6%

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

      1. associate-/l* 94.5%

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

      2. associate-/r/ 91.2%

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

   6. Simplified 91.2%

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

   if -1 < y < 1

   1. Initial program 93.0%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 98.7%

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

4. Final simplification 94.8%

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

Alternative 6: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (- (/ y (/ z x)) x) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = (y / (z / x)) - x;
	} 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 <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = (y / (z / x)) - x
    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 <= -1.0) || !(y <= 1.0)) {
		tmp = (y / (z / x)) - x;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = (y / (z / x)) - x
	else:
		tmp = (x / z) - x
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 87.1%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in y around inf 91.6%

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

      1. associate-/l* 94.5%

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

   6. Simplified 94.5%

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

   if -1 < y < 1

   1. Initial program 93.0%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 98.7%

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

4. Final simplification 96.5%

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

Alternative 7: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -60000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -60000000.0) (not (<= z 1.0)))
   (- (* x (/ y z)) x)
   (/ (* x (+ y 1.0)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -60000000.0) || !(z <= 1.0)) {
		tmp = (x * (y / z)) - x;
	} 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 <= (-60000000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x * (y / z)) - x
    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 <= -60000000.0) || !(z <= 1.0)) {
		tmp = (x * (y / z)) - x;
	} else {
		tmp = (x * (y + 1.0)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -60000000.0) or not (z <= 1.0):
		tmp = (x * (y / z)) - x
	else:
		tmp = (x * (y + 1.0)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -60000000.0) || !(z <= 1.0))
		tmp = Float64(Float64(x * Float64(y / z)) - x);
	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 <= -60000000.0) || ~((z <= 1.0)))
		tmp = (x * (y / z)) - x;
	else
		tmp = (x * (y + 1.0)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -60000000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6e7 or 1 < z

   1. Initial program 81.6%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in y around inf 91.9%

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

      1. associate-/l* 94.7%

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

      2. associate-/r/ 99.5%

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

   6. Simplified 99.5%

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

   if -6e7 < z < 1

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -60000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \end{array} \]

Alternative 8: 95.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = (x * (y / z)) - x
    else if (y <= 1.0d0) then
        tmp = (x / z) - x
    else
        tmp = (x / (z / y)) - x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 82.9%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in y around inf 89.0%

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

      1. associate-/l* 96.2%

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

      2. associate-/r/ 90.9%

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

   6. Simplified 90.9%

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

   if -1 < y < 1

   1. Initial program 93.0%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 98.7%

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

   if 1 < y

   1. Initial program 91.4%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in y around inf 94.1%

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

      1. associate-/l* 92.8%

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

      2. associate-/r/ 91.4%

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

   6. Simplified 91.4%

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

      1. *-commutative 91.4%

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

      2. clear-num 91.4%

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

      3. un-div-inv 91.5%

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

   8. Applied egg-rr 91.5%

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

4. Final simplification 94.8%

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

Alternative 9: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+56} \lor \neg \left(y \leq 1.65 \cdot 10^{+63}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.7e+56) (not (<= y 1.65e+63))) (* y (/ x z)) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.7e+56) || !(y <= 1.65e+63)) {
		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 <= (-3.7d+56)) .or. (.not. (y <= 1.65d+63))) 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 <= -3.7e+56) || !(y <= 1.65e+63)) {
		tmp = y * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.7e+56) or not (y <= 1.65e+63):
		tmp = y * (x / z)
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.7e+56) || !(y <= 1.65e+63))
		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 <= -3.7e+56) || ~((y <= 1.65e+63)))
		tmp = y * (x / z);
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.7e+56], N[Not[LessEqual[y, 1.65e+63]], $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 < -3.69999999999999997e56 or 1.6500000000000001e63 < y

   1. Initial program 87.3%

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

      1. associate-/l* 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in y around inf 70.0%

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

      1. associate-/r/ 75.1%

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

   6. Applied egg-rr 75.1%

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

   if -3.69999999999999997e56 < y < 1.6500000000000001e63

   1. Initial program 91.6%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 93.6%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+56} \lor \neg \left(y \leq 1.65 \cdot 10^{+63}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 10: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.5e+58)
   (/ y (/ z x))
   (if (<= y 1.5e+63) (- (/ x z) x) (* y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e+58) {
		tmp = y / (z / x);
	} else if (y <= 1.5e+63) {
		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 <= (-1.5d+58)) then
        tmp = y / (z / x)
    else if (y <= 1.5d+63) 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 <= -1.5e+58) {
		tmp = y / (z / x);
	} else if (y <= 1.5e+63) {
		tmp = (x / z) - x;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.5e+58:
		tmp = y / (z / x)
	elif y <= 1.5e+63:
		tmp = (x / z) - x
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.5e+58)
		tmp = Float64(y / Float64(z / x));
	elseif (y <= 1.5e+63)
		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 <= -1.5e+58)
		tmp = y / (z / x);
	elseif (y <= 1.5e+63)
		tmp = (x / z) - x;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.5e+58], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+63], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5000000000000001e58

   1. Initial program 84.1%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in x around 0 88.3%

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

   5. Taylor expanded in y around inf 64.2%

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

      1. associate-/r/ 73.4%

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

   7. Applied egg-rr 73.4%

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

   if -1.5000000000000001e58 < y < 1.5e63

   1. Initial program 91.6%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 93.6%

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

   if 1.5e63 < y

   1. Initial program 90.4%

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

      1. associate-/l* 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in y around inf 75.6%

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

      1. associate-/r/ 76.8%

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

   6. Applied egg-rr 76.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 11: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+58}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{+106}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.12e+58)
   (/ y (/ z x))
   (if (<= y 1.08e+106) (- (/ x z) x) (/ (* x y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.12e+58) {
		tmp = y / (z / x);
	} else if (y <= 1.08e+106) {
		tmp = (x / z) - x;
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.12d+58)) then
        tmp = y / (z / x)
    else if (y <= 1.08d+106) then
        tmp = (x / z) - x
    else
        tmp = (x * y) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.12e+58) {
		tmp = y / (z / x);
	} else if (y <= 1.08e+106) {
		tmp = (x / z) - x;
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.12e+58:
		tmp = y / (z / x)
	elif y <= 1.08e+106:
		tmp = (x / z) - x
	else:
		tmp = (x * y) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.12e+58)
		tmp = Float64(y / Float64(z / x));
	elseif (y <= 1.08e+106)
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = Float64(Float64(x * y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.12e+58)
		tmp = y / (z / x);
	elseif (y <= 1.08e+106)
		tmp = (x / z) - x;
	else
		tmp = (x * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.12e+58], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.08e+106], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.12e58

   1. Initial program 84.1%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in x around 0 88.3%

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

   5. Taylor expanded in y around inf 64.2%

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

      1. associate-/r/ 73.4%

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

   7. Applied egg-rr 73.4%

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

   if -1.12e58 < y < 1.08000000000000007e106

   1. Initial program 90.4%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in y around 0 91.2%

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

   if 1.08000000000000007e106 < y

   1. Initial program 95.1%

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

   2. Taylor expanded in y around inf 89.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+58}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{+106}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternative 12: 64.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 750000000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0) (- x) (if (<= z 750000000000.0) (/ x z) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = -x;
	} else if (z <= 750000000000.0) {
		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 <= 750000000000.0d0) 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 <= 750000000000.0) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = -x
	elif z <= 750000000000.0:
		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 <= 750000000000.0)
		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 <= 750000000000.0)
		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, 750000000000.0], N[(x / z), $MachinePrecision], (-x)]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 7.5e11 < z

   1. Initial program 81.5%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in z around inf 75.0%

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

      1. neg-mul-1 75.0%

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

   6. Simplified 75.0%

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

   if -1 < z < 7.5e11

   1. Initial program 99.8%

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

      1. associate-/l* 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in y around 0 58.5%

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

      1. associate-/l* 58.4%

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

      2. associate-/r/ 58.4%

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

   6. Simplified 58.4%

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

   7. Taylor expanded in z around 0 57.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 750000000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 13: 39.5% 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 89.9%

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

3. Simplified 95.7%

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

4. Taylor expanded in z around inf 41.9%

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

   1. neg-mul-1 41.9%

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

6. Simplified 41.9%

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

7. Final simplification 41.9%

   \[\leadsto -x \]

Developer target: 99.4% 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 2023196 
(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))