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

Percentage Accurate: 87.8% → 99.3%

Time: 7.5s

Alternatives: 12

Speedup: 1.0×

• 20.8% 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: 87.8% 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: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.4 \lor \neg \left(z \leq 3.8 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y - \left(z + -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.4) (not (<= z 3.8e-161)))
   (/ x (/ z (- y (+ z -1.0))))
   (/ (* x (+ 1.0 (- y z))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.4) || !(z <= 3.8e-161)) {
		tmp = x / (z / (y - (z + -1.0)));
	} else {
		tmp = (x * (1.0 + (y - z))) / 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 <= (-0.4d0)) .or. (.not. (z <= 3.8d-161))) then
        tmp = x / (z / (y - (z + (-1.0d0))))
    else
        tmp = (x * (1.0d0 + (y - z))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.4) || !(z <= 3.8e-161)) {
		tmp = x / (z / (y - (z + -1.0)));
	} else {
		tmp = (x * (1.0 + (y - z))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -0.4) or not (z <= 3.8e-161):
		tmp = x / (z / (y - (z + -1.0)))
	else:
		tmp = (x * (1.0 + (y - z))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.4) || !(z <= 3.8e-161))
		tmp = Float64(x / Float64(z / Float64(y - Float64(z + -1.0))));
	else
		tmp = Float64(Float64(x * Float64(1.0 + Float64(y - z))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -0.4) || ~((z <= 3.8e-161)))
		tmp = x / (z / (y - (z + -1.0)));
	else
		tmp = (x * (1.0 + (y - z))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.4], N[Not[LessEqual[z, 3.8e-161]], $MachinePrecision]], N[(x / N[(z / N[(y - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.40000000000000002 or 3.8000000000000001e-161 < z

   1. Initial program 86.6%

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

      1. associate-/l* 99.9%

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

      2. associate-/r/ 75.8%

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

   3. Applied egg-rr 75.8%

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

      1. associate-*l/ 86.6%

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

      2. associate-/l* 99.9%

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

      3. associate-+l- 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -0.40000000000000002 < z < 3.8000000000000001e-161

   1. Initial program 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.4 \lor \neg \left(z \leq 3.8 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y - \left(z + -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \end{array} \]

Alternative 2: 66.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+17}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{-218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-227}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))))
   (if (<= z -4.5e+17)
     (- x)
     (if (<= z -2.45e-218)
       t_0
       (if (<= z 7e-227)
         (/ x z)
         (if (<= z 7e-116) t_0 (if (<= z 4.2e-6) (/ x z) (- x))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (z <= -4.5e+17) {
		tmp = -x;
	} else if (z <= -2.45e-218) {
		tmp = t_0;
	} else if (z <= 7e-227) {
		tmp = x / z;
	} else if (z <= 7e-116) {
		tmp = t_0;
	} else if (z <= 4.2e-6) {
		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) :: t_0
    real(8) :: tmp
    t_0 = y * (x / z)
    if (z <= (-4.5d+17)) then
        tmp = -x
    else if (z <= (-2.45d-218)) then
        tmp = t_0
    else if (z <= 7d-227) then
        tmp = x / z
    else if (z <= 7d-116) then
        tmp = t_0
    else if (z <= 4.2d-6) 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 t_0 = y * (x / z);
	double tmp;
	if (z <= -4.5e+17) {
		tmp = -x;
	} else if (z <= -2.45e-218) {
		tmp = t_0;
	} else if (z <= 7e-227) {
		tmp = x / z;
	} else if (z <= 7e-116) {
		tmp = t_0;
	} else if (z <= 4.2e-6) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / z)
	tmp = 0
	if z <= -4.5e+17:
		tmp = -x
	elif z <= -2.45e-218:
		tmp = t_0
	elif z <= 7e-227:
		tmp = x / z
	elif z <= 7e-116:
		tmp = t_0
	elif z <= 4.2e-6:
		tmp = x / z
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (z <= -4.5e+17)
		tmp = Float64(-x);
	elseif (z <= -2.45e-218)
		tmp = t_0;
	elseif (z <= 7e-227)
		tmp = Float64(x / z);
	elseif (z <= 7e-116)
		tmp = t_0;
	elseif (z <= 4.2e-6)
		tmp = Float64(x / z);
	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 <= -4.5e+17)
		tmp = -x;
	elseif (z <= -2.45e-218)
		tmp = t_0;
	elseif (z <= 7e-227)
		tmp = x / z;
	elseif (z <= 7e-116)
		tmp = t_0;
	elseif (z <= 4.2e-6)
		tmp = x / z;
	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, -4.5e+17], (-x), If[LessEqual[z, -2.45e-218], t$95$0, If[LessEqual[z, 7e-227], N[(x / z), $MachinePrecision], If[LessEqual[z, 7e-116], t$95$0, If[LessEqual[z, 4.2e-6], N[(x / z), $MachinePrecision], (-x)]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.5e17 or 4.1999999999999996e-6 < z

   1. Initial program 82.9%

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

   2. Taylor expanded in z around inf 76.4%

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

      1. neg-mul-1 76.4%

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

   4. Simplified 76.4%

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

   if -4.5e17 < z < -2.44999999999999989e-218 or 7.0000000000000002e-227 < z < 6.99999999999999968e-116

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 63.3%

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

      1. associate-/l* 65.8%

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

      2. div-inv 67.0%

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

      3. clear-num 67.0%

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

   4. Applied egg-rr 67.0%

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

   if -2.44999999999999989e-218 < z < 7.0000000000000002e-227 or 6.99999999999999968e-116 < z < 4.1999999999999996e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 100.0%

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

   3. Taylor expanded in y around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. mul-1-neg 69.3%

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

      3. unsub-neg 69.3%

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

   5. Simplified 69.3%

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

   6. Taylor expanded in z around 0 68.3%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+17}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{-218}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-227}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-116}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 3: 95.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+18} \lor \neg \left(z \leq 5.5 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{y \cdot x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(y - z\right)\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7.5e+18) (not (<= z 5.5e+113)))
   (- (/ (* y x) z) x)
   (* (+ 1.0 (- y z)) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.5e+18) || !(z <= 5.5e+113)) {
		tmp = ((y * x) / z) - x;
	} else {
		tmp = (1.0 + (y - z)) * (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 ((z <= (-7.5d+18)) .or. (.not. (z <= 5.5d+113))) then
        tmp = ((y * x) / z) - x
    else
        tmp = (1.0d0 + (y - z)) * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.5e+18) || !(z <= 5.5e+113)) {
		tmp = ((y * x) / z) - x;
	} else {
		tmp = (1.0 + (y - z)) * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -7.5e+18) or not (z <= 5.5e+113):
		tmp = ((y * x) / z) - x
	else:
		tmp = (1.0 + (y - z)) * (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -7.5e+18) || !(z <= 5.5e+113))
		tmp = Float64(Float64(Float64(y * x) / z) - x);
	else
		tmp = Float64(Float64(1.0 + Float64(y - z)) * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -7.5e+18) || ~((z <= 5.5e+113)))
		tmp = ((y * x) / z) - x;
	else
		tmp = (1.0 + (y - z)) * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -7.5e+18], N[Not[LessEqual[z, 5.5e+113]], $MachinePrecision]], N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], N[(N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.5e18 or 5.5000000000000001e113 < z

   1. Initial program 81.6%

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

   2. Taylor expanded in z around inf 95.9%

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

   3. Taylor expanded in y around inf 95.9%

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

   if -7.5e18 < z < 5.5000000000000001e113

   1. Initial program 98.3%

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

      1. associate-/l* 89.5%

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

      2. associate-/r/ 99.3%

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

   3. Applied egg-rr 99.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+18} \lor \neg \left(z \leq 5.5 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{y \cdot x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(y - z\right)\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{-16} \lor \neg \left(z \leq 5 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y - \left(z + -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.6e-16) (not (<= z 5e-161)))
   (/ x (/ z (- y (+ z -1.0))))
   (/ (+ x (* y x)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.6e-16) || !(z <= 5e-161)) {
		tmp = x / (z / (y - (z + -1.0)));
	} else {
		tmp = (x + (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 ((z <= (-6.6d-16)) .or. (.not. (z <= 5d-161))) then
        tmp = x / (z / (y - (z + (-1.0d0))))
    else
        tmp = (x + (y * x)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.6e-16) || !(z <= 5e-161)) {
		tmp = x / (z / (y - (z + -1.0)));
	} else {
		tmp = (x + (y * x)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.6e-16) or not (z <= 5e-161):
		tmp = x / (z / (y - (z + -1.0)))
	else:
		tmp = (x + (y * x)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.6e-16) || !(z <= 5e-161))
		tmp = Float64(x / Float64(z / Float64(y - Float64(z + -1.0))));
	else
		tmp = Float64(Float64(x + Float64(y * x)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.6e-16) || ~((z <= 5e-161)))
		tmp = x / (z / (y - (z + -1.0)));
	else
		tmp = (x + (y * x)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.6e-16], N[Not[LessEqual[z, 5e-161]], $MachinePrecision]], N[(x / N[(z / N[(y - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.59999999999999976e-16 or 4.9999999999999999e-161 < z

   1. Initial program 86.8%

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

      1. associate-/l* 99.9%

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

      2. associate-/r/ 76.2%

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

   3. Applied egg-rr 76.2%

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

      1. associate-*l/ 86.8%

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

      2. associate-/l* 99.9%

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

      3. associate-+l- 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -6.59999999999999976e-16 < z < 4.9999999999999999e-161

   1. Initial program 99.9%

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

      1. distribute-lft-in 100.0%

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

      2. fma-def 100.0%

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

      3. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{-16} \lor \neg \left(z \leq 5 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y - \left(z + -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot x}{z}\\ \end{array} \]

Alternative 5: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+142}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+132}:\\ \;\;\;\;\left(1 + \left(y - z\right)\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.5e+142)
   (- x)
   (if (<= z 4.6e+132) (* (+ 1.0 (- y z)) (/ x z)) (- (/ x z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e+142) {
		tmp = -x;
	} else if (z <= 4.6e+132) {
		tmp = (1.0 + (y - z)) * (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 (z <= (-7.5d+142)) then
        tmp = -x
    else if (z <= 4.6d+132) then
        tmp = (1.0d0 + (y - z)) * (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 (z <= -7.5e+142) {
		tmp = -x;
	} else if (z <= 4.6e+132) {
		tmp = (1.0 + (y - z)) * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.5e+142:
		tmp = -x
	elif z <= 4.6e+132:
		tmp = (1.0 + (y - z)) * (x / z)
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.5e+142)
		tmp = Float64(-x);
	elseif (z <= 4.6e+132)
		tmp = Float64(Float64(1.0 + Float64(y - z)) * 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 (z <= -7.5e+142)
		tmp = -x;
	elseif (z <= 4.6e+132)
		tmp = (1.0 + (y - z)) * (x / z);
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.5e+142], (-x), If[LessEqual[z, 4.6e+132], N[(N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.5000000000000002e142

   1. Initial program 70.1%

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

   2. Taylor expanded in z around inf 90.3%

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

      1. neg-mul-1 90.3%

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

   4. Simplified 90.3%

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

   if -7.5000000000000002e142 < z < 4.6000000000000003e132

   1. Initial program 98.0%

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

      1. associate-/l* 90.9%

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

      2. associate-/r/ 97.0%

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

   3. Applied egg-rr 97.0%

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

   if 4.6000000000000003e132 < z

   1. Initial program 84.1%

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

   2. Taylor expanded in z around inf 96.9%

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

   3. Taylor expanded in y around 0 88.0%

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

      1. +-commutative 88.0%

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

      2. mul-1-neg 88.0%

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

      3. unsub-neg 88.0%

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

   5. Simplified 88.0%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+142}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+132}:\\ \;\;\;\;\left(1 + \left(y - z\right)\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 6: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -160 \lor \neg \left(y \leq 9.8 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{x + y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -160.0) (not (<= y 9.8e+55)))
   (/ (+ x (* y x)) z)
   (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -160.0) || !(y <= 9.8e+55)) {
		tmp = (x + (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 <= (-160.0d0)) .or. (.not. (y <= 9.8d+55))) then
        tmp = (x + (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 <= -160.0) || !(y <= 9.8e+55)) {
		tmp = (x + (y * x)) / z;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -160.0) or not (y <= 9.8e+55):
		tmp = (x + (y * x)) / z
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -160.0) || !(y <= 9.8e+55))
		tmp = Float64(Float64(x + Float64(y * x)) / z);
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -160.0) || ~((y <= 9.8e+55)))
		tmp = (x + (y * x)) / z;
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -160.0], N[Not[LessEqual[y, 9.8e+55]], $MachinePrecision]], N[(N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -160 or 9.80000000000000029e55 < y

   1. Initial program 94.2%

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

      1. distribute-lft-in 94.2%

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

      2. fma-def 94.2%

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

      3. *-rgt-identity 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in z around 0 80.9%

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

   if -160 < y < 9.80000000000000029e55

   1. Initial program 90.6%

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

   2. Taylor expanded in z around inf 98.6%

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

   3. Taylor expanded in y around 0 97.0%

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

      1. +-commutative 97.0%

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

      2. mul-1-neg 97.0%

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

      3. unsub-neg 97.0%

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

   5. Simplified 97.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -160 \lor \neg \left(y \leq 9.8 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{x + y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 7: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -14000 \lor \neg \left(y \leq 1.7 \cdot 10^{+53}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -14000.0) (not (<= y 1.7e+53))) (* y (/ x z)) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -14000.0) || !(y <= 1.7e+53)) {
		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 <= (-14000.0d0)) .or. (.not. (y <= 1.7d+53))) 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 <= -14000.0) || !(y <= 1.7e+53)) {
		tmp = y * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -14000.0) or not (y <= 1.7e+53):
		tmp = y * (x / z)
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -14000.0) || !(y <= 1.7e+53))
		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 <= -14000.0) || ~((y <= 1.7e+53)))
		tmp = y * (x / z);
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -14000.0], N[Not[LessEqual[y, 1.7e+53]], $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 < -14000 or 1.69999999999999999e53 < y

   1. Initial program 94.0%

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

   2. Taylor expanded in y around inf 81.5%

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

      1. associate-/l* 78.1%

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

      2. div-inv 78.1%

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

      3. clear-num 78.4%

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

   4. Applied egg-rr 78.4%

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

   if -14000 < y < 1.69999999999999999e53

   1. Initial program 90.9%

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

   2. Taylor expanded in z around inf 98.7%

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

   3. Taylor expanded in y around 0 95.7%

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

      1. +-commutative 95.7%

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

      2. mul-1-neg 95.7%

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

      3. unsub-neg 95.7%

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

   5. Simplified 95.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -14000 \lor \neg \left(y \leq 1.7 \cdot 10^{+53}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 8: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -11000 \lor \neg \left(y \leq 1.5 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -11000.0) (not (<= y 1.5e+54))) (/ (* y x) z) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -11000.0) || !(y <= 1.5e+54)) {
		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 <= (-11000.0d0)) .or. (.not. (y <= 1.5d+54))) 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 <= -11000.0) || !(y <= 1.5e+54)) {
		tmp = (y * x) / z;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -11000.0) or not (y <= 1.5e+54):
		tmp = (y * x) / z
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -11000.0) || !(y <= 1.5e+54))
		tmp = Float64(Float64(y * x) / z);
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -11000.0) || ~((y <= 1.5e+54)))
		tmp = (y * x) / z;
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -11000.0], N[Not[LessEqual[y, 1.5e+54]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -11000 or 1.4999999999999999e54 < y

   1. Initial program 94.0%

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

   2. Taylor expanded in y around inf 81.5%

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

   if -11000 < y < 1.4999999999999999e54

   1. Initial program 90.9%

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

   2. Taylor expanded in z around inf 98.7%

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

   3. Taylor expanded in y around 0 95.7%

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

      1. +-commutative 95.7%

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

      2. mul-1-neg 95.7%

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

      3. unsub-neg 95.7%

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

   5. Simplified 95.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11000 \lor \neg \left(y \leq 1.5 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 9: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

2. Taylor expanded in z around inf 97.7%

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

3. Final simplification 97.7%

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

Alternative 10: 65.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -32000000000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -32000000000.0) (- x) (if (<= z 4.2e-6) (/ x z) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -32000000000.0) {
		tmp = -x;
	} else if (z <= 4.2e-6) {
		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 <= (-32000000000.0d0)) then
        tmp = -x
    else if (z <= 4.2d-6) 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 <= -32000000000.0) {
		tmp = -x;
	} else if (z <= 4.2e-6) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -32000000000.0:
		tmp = -x
	elif z <= 4.2e-6:
		tmp = x / z
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -32000000000.0)
		tmp = Float64(-x);
	elseif (z <= 4.2e-6)
		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 <= -32000000000.0)
		tmp = -x;
	elseif (z <= 4.2e-6)
		tmp = x / z;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -32000000000.0], (-x), If[LessEqual[z, 4.2e-6], N[(x / z), $MachinePrecision], (-x)]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2e10 or 4.1999999999999996e-6 < z

   1. Initial program 83.2%

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

   2. Taylor expanded in z around inf 75.9%

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

      1. neg-mul-1 75.9%

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

   4. Simplified 75.9%

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

   if -3.2e10 < z < 4.1999999999999996e-6

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 99.9%

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

   3. Taylor expanded in y around 0 57.5%

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

      1. +-commutative 57.5%

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

      2. mul-1-neg 57.5%

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

      3. unsub-neg 57.5%

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

   5. Simplified 57.5%

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

   6. Taylor expanded in z around 0 56.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -32000000000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 11: 39.7% 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 92.2%

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

2. Taylor expanded in z around inf 36.7%

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

   1. neg-mul-1 36.7%

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

4. Simplified 36.7%

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

5. Final simplification 36.7%

   \[\leadsto -x \]

Alternative 12: 3.0% accurate, 9.0× 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 x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 92.2%

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

2. Taylor expanded in z around inf 31.5%

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

   1. mul-1-neg 31.5%

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

   2. distribute-rgt-neg-in 31.5%

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

4. Simplified 31.5%

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

   1. expm1-log1p-u 26.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{z \cdot \left(-x\right)}{z}\right)\right)} \]

   2. expm1-udef 7.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{z \cdot \left(-x\right)}{z}\right)} - 1} \]

   3. div-inv 7.9%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(z \cdot \left(-x\right)\right) \cdot \frac{1}{z}}\right)} - 1 \]

   4. associate-*l* 12.6%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{z \cdot \left(\left(-x\right) \cdot \frac{1}{z}\right)}\right)} - 1 \]

   5. add-sqr-sqrt 11.1%

      \[\leadsto e^{\mathsf{log1p}\left(z \cdot \left(\color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \frac{1}{z}\right)\right)} - 1 \]

   6. sqrt-unprod 11.7%

      \[\leadsto e^{\mathsf{log1p}\left(z \cdot \left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \cdot \frac{1}{z}\right)\right)} - 1 \]

   7. sqr-neg 11.7%

      \[\leadsto e^{\mathsf{log1p}\left(z \cdot \left(\sqrt{\color{blue}{x \cdot x}} \cdot \frac{1}{z}\right)\right)} - 1 \]

   8. sqrt-unprod 4.1%

      \[\leadsto e^{\mathsf{log1p}\left(z \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{z}\right)\right)} - 1 \]

   9. add-sqr-sqrt 5.0%

      \[\leadsto e^{\mathsf{log1p}\left(z \cdot \left(\color{blue}{x} \cdot \frac{1}{z}\right)\right)} - 1 \]

   10. div-inv 5.0%

       \[\leadsto e^{\mathsf{log1p}\left(z \cdot \color{blue}{\frac{x}{z}}\right)} - 1 \]

6. Applied egg-rr 5.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(z \cdot \frac{x}{z}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 5.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(z \cdot \frac{x}{z}\right)\right)} \]

   2. expm1-log1p 8.9%

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

   3. associate-*r/ 2.8%

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

   4. associate-*l/ 2.9%

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

   5. *-inverses 2.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   6. *-lft-identity 2.9%

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

8. Simplified 2.9%

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

9. Final simplification 2.9%

   \[\leadsto x \]

Developer target: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 + y) * (x / z)) - x
    if (x < (-2.71483106713436d-162)) then
        tmp = t_0
    else if (x < 3.874108816439546d-197) then
        tmp = (x * ((y - z) + 1.0d0)) * (1.0d0 / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023279 
(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))