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

Percentage Accurate: 88.5% → 98.9%

Time: 6.2s

Alternatives: 11

Speedup: 0.7×

• 20.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% 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: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+108} \lor \neg \left(z \leq 1.5 \cdot 10^{+64}\right):\\ \;\;\;\;\frac{\left(y - z\right) + 1}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2e+108) (not (<= z 1.5e+64)))
   (* (/ (+ (- y z) 1.0) z) x)
   (/ (fma x (- y z) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2e+108) || !(z <= 1.5e+64)) {
		tmp = (((y - z) + 1.0) / z) * x;
	} else {
		tmp = fma(x, (y - z), x) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2e+108) || !(z <= 1.5e+64))
		tmp = Float64(Float64(Float64(Float64(y - z) + 1.0) / z) * x);
	else
		tmp = Float64(fma(x, Float64(y - z), x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2e+108], N[Not[LessEqual[z, 1.5e+64]], $MachinePrecision]], N[(N[(N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[(y - z), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0000000000000001e108 or 1.5000000000000001e64 < z

   1. Initial program 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-/l* 58.1%

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

      3. associate-/r/ 100.0%

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

   3. Applied egg-rr 100.0%

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

   if -2.0000000000000001e108 < z < 1.5000000000000001e64

   1. Initial program 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. fma-def 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+108} \lor \neg \left(z \leq 1.5 \cdot 10^{+64}\right):\\ \;\;\;\;\frac{\left(y - z\right) + 1}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \end{array} \]

Alternative 2: 64.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -9.8 \cdot 10^{+29}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -0.0072:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-305}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))))
   (if (<= z -9.8e+29)
     (- x)
     (if (<= z -0.0072)
       t_0
       (if (<= z -1.9e-180)
         (/ x z)
         (if (<= z -1.12e-305)
           t_0
           (if (<= z 2.8e-44) (/ x z) (if (<= z 6.2e+65) t_0 (- x)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (z <= -9.8e+29) {
		tmp = -x;
	} else if (z <= -0.0072) {
		tmp = t_0;
	} else if (z <= -1.9e-180) {
		tmp = x / z;
	} else if (z <= -1.12e-305) {
		tmp = t_0;
	} else if (z <= 2.8e-44) {
		tmp = x / z;
	} else if (z <= 6.2e+65) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x / z)
    if (z <= (-9.8d+29)) then
        tmp = -x
    else if (z <= (-0.0072d0)) then
        tmp = t_0
    else if (z <= (-1.9d-180)) then
        tmp = x / z
    else if (z <= (-1.12d-305)) then
        tmp = t_0
    else if (z <= 2.8d-44) then
        tmp = x / z
    else if (z <= 6.2d+65) then
        tmp = t_0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (z <= -9.8e+29) {
		tmp = -x;
	} else if (z <= -0.0072) {
		tmp = t_0;
	} else if (z <= -1.9e-180) {
		tmp = x / z;
	} else if (z <= -1.12e-305) {
		tmp = t_0;
	} else if (z <= 2.8e-44) {
		tmp = x / z;
	} else if (z <= 6.2e+65) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / z)
	tmp = 0
	if z <= -9.8e+29:
		tmp = -x
	elif z <= -0.0072:
		tmp = t_0
	elif z <= -1.9e-180:
		tmp = x / z
	elif z <= -1.12e-305:
		tmp = t_0
	elif z <= 2.8e-44:
		tmp = x / z
	elif z <= 6.2e+65:
		tmp = t_0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (z <= -9.8e+29)
		tmp = Float64(-x);
	elseif (z <= -0.0072)
		tmp = t_0;
	elseif (z <= -1.9e-180)
		tmp = Float64(x / z);
	elseif (z <= -1.12e-305)
		tmp = t_0;
	elseif (z <= 2.8e-44)
		tmp = Float64(x / z);
	elseif (z <= 6.2e+65)
		tmp = t_0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x / z);
	tmp = 0.0;
	if (z <= -9.8e+29)
		tmp = -x;
	elseif (z <= -0.0072)
		tmp = t_0;
	elseif (z <= -1.9e-180)
		tmp = x / z;
	elseif (z <= -1.12e-305)
		tmp = t_0;
	elseif (z <= 2.8e-44)
		tmp = x / z;
	elseif (z <= 6.2e+65)
		tmp = t_0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.8e+29], (-x), If[LessEqual[z, -0.0072], t$95$0, If[LessEqual[z, -1.9e-180], N[(x / z), $MachinePrecision], If[LessEqual[z, -1.12e-305], t$95$0, If[LessEqual[z, 2.8e-44], N[(x / z), $MachinePrecision], If[LessEqual[z, 6.2e+65], t$95$0, (-x)]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.8000000000000003e29 or 6.19999999999999981e65 < z

   1. Initial program 84.9%

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

   2. Taylor expanded in z around inf 77.4%

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

      1. mul-1-neg 77.4%

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

   4. Simplified 77.4%

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

   if -9.8000000000000003e29 < z < -0.0071999999999999998 or -1.9e-180 < z < -1.1200000000000001e-305 or 2.8e-44 < z < 6.19999999999999981e65

   1. Initial program 98.3%

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

   2. Taylor expanded in y around inf 67.2%

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

      1. associate-*l/ 70.1%

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

      2. *-commutative 70.1%

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

   4. Simplified 70.1%

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

   if -0.0071999999999999998 < z < -1.9e-180 or -1.1200000000000001e-305 < z < 2.8e-44

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 98.6%

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

   3. Taylor expanded in y around 0 64.6%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+29}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -0.0072:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-305}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+65}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 3: 62.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+131}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -0.0072:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-179}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-300}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))))
   (if (<= z -9.5e+131)
     (- x)
     (if (<= z -0.0072)
       (* x (/ y z))
       (if (<= z -5.1e-179)
         (/ x z)
         (if (<= z -2.1e-300)
           t_0
           (if (<= z 8.5e-43) (/ x z) (if (<= z 6.5e+65) t_0 (- x)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (z <= -9.5e+131) {
		tmp = -x;
	} else if (z <= -0.0072) {
		tmp = x * (y / z);
	} else if (z <= -5.1e-179) {
		tmp = x / z;
	} else if (z <= -2.1e-300) {
		tmp = t_0;
	} else if (z <= 8.5e-43) {
		tmp = x / z;
	} else if (z <= 6.5e+65) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x / z)
    if (z <= (-9.5d+131)) then
        tmp = -x
    else if (z <= (-0.0072d0)) then
        tmp = x * (y / z)
    else if (z <= (-5.1d-179)) then
        tmp = x / z
    else if (z <= (-2.1d-300)) then
        tmp = t_0
    else if (z <= 8.5d-43) then
        tmp = x / z
    else if (z <= 6.5d+65) then
        tmp = t_0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (z <= -9.5e+131) {
		tmp = -x;
	} else if (z <= -0.0072) {
		tmp = x * (y / z);
	} else if (z <= -5.1e-179) {
		tmp = x / z;
	} else if (z <= -2.1e-300) {
		tmp = t_0;
	} else if (z <= 8.5e-43) {
		tmp = x / z;
	} else if (z <= 6.5e+65) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / z)
	tmp = 0
	if z <= -9.5e+131:
		tmp = -x
	elif z <= -0.0072:
		tmp = x * (y / z)
	elif z <= -5.1e-179:
		tmp = x / z
	elif z <= -2.1e-300:
		tmp = t_0
	elif z <= 8.5e-43:
		tmp = x / z
	elif z <= 6.5e+65:
		tmp = t_0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (z <= -9.5e+131)
		tmp = Float64(-x);
	elseif (z <= -0.0072)
		tmp = Float64(x * Float64(y / z));
	elseif (z <= -5.1e-179)
		tmp = Float64(x / z);
	elseif (z <= -2.1e-300)
		tmp = t_0;
	elseif (z <= 8.5e-43)
		tmp = Float64(x / z);
	elseif (z <= 6.5e+65)
		tmp = t_0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x / z);
	tmp = 0.0;
	if (z <= -9.5e+131)
		tmp = -x;
	elseif (z <= -0.0072)
		tmp = x * (y / z);
	elseif (z <= -5.1e-179)
		tmp = x / z;
	elseif (z <= -2.1e-300)
		tmp = t_0;
	elseif (z <= 8.5e-43)
		tmp = x / z;
	elseif (z <= 6.5e+65)
		tmp = t_0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.5e+131], (-x), If[LessEqual[z, -0.0072], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -5.1e-179], N[(x / z), $MachinePrecision], If[LessEqual[z, -2.1e-300], t$95$0, If[LessEqual[z, 8.5e-43], N[(x / z), $MachinePrecision], If[LessEqual[z, 6.5e+65], t$95$0, (-x)]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9.50000000000000015e131 or 6.5000000000000003e65 < z

   1. Initial program 79.3%

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

   2. Taylor expanded in z around inf 89.5%

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

      1. mul-1-neg 89.5%

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

   4. Simplified 89.5%

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

   if -9.50000000000000015e131 < z < -0.0071999999999999998

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 82.9%

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

      3. associate-/r/ 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around inf 63.0%

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

   if -0.0071999999999999998 < z < -5.10000000000000028e-179 or -2.10000000000000004e-300 < z < 8.50000000000000056e-43

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 98.6%

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

   3. Taylor expanded in y around 0 64.6%

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

   if -5.10000000000000028e-179 < z < -2.10000000000000004e-300 or 8.50000000000000056e-43 < z < 6.5000000000000003e65

   1. Initial program 98.0%

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

   2. Taylor expanded in y around inf 65.8%

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

      1. associate-*l/ 69.3%

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

      2. *-commutative 69.3%

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

   4. Simplified 69.3%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+131}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -0.0072:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-179}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-300}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+65}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{z} - x\\ t_1 := \frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 50000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ x z) x)) (t_1 (/ (* x (+ y 1.0)) z)))
   (if (<= y -1.1e-8)
     t_1
     (if (<= y 50000.0)
       t_0
       (if (<= y 3.4e+31) t_1 (if (<= y 8.6e+84) t_0 (/ (* y x) z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x / z) - x;
	double t_1 = (x * (y + 1.0)) / z;
	double tmp;
	if (y <= -1.1e-8) {
		tmp = t_1;
	} else if (y <= 50000.0) {
		tmp = t_0;
	} else if (y <= 3.4e+31) {
		tmp = t_1;
	} else if (y <= 8.6e+84) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x / z) - x
    t_1 = (x * (y + 1.0d0)) / z
    if (y <= (-1.1d-8)) then
        tmp = t_1
    else if (y <= 50000.0d0) then
        tmp = t_0
    else if (y <= 3.4d+31) then
        tmp = t_1
    else if (y <= 8.6d+84) then
        tmp = t_0
    else
        tmp = (y * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x / z) - x;
	double t_1 = (x * (y + 1.0)) / z;
	double tmp;
	if (y <= -1.1e-8) {
		tmp = t_1;
	} else if (y <= 50000.0) {
		tmp = t_0;
	} else if (y <= 3.4e+31) {
		tmp = t_1;
	} else if (y <= 8.6e+84) {
		tmp = t_0;
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / z) - x
	t_1 = (x * (y + 1.0)) / z
	tmp = 0
	if y <= -1.1e-8:
		tmp = t_1
	elif y <= 50000.0:
		tmp = t_0
	elif y <= 3.4e+31:
		tmp = t_1
	elif y <= 8.6e+84:
		tmp = t_0
	else:
		tmp = (y * x) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / z) - x)
	t_1 = Float64(Float64(x * Float64(y + 1.0)) / z)
	tmp = 0.0
	if (y <= -1.1e-8)
		tmp = t_1;
	elseif (y <= 50000.0)
		tmp = t_0;
	elseif (y <= 3.4e+31)
		tmp = t_1;
	elseif (y <= 8.6e+84)
		tmp = t_0;
	else
		tmp = Float64(Float64(y * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / z) - x;
	t_1 = (x * (y + 1.0)) / z;
	tmp = 0.0;
	if (y <= -1.1e-8)
		tmp = t_1;
	elseif (y <= 50000.0)
		tmp = t_0;
	elseif (y <= 3.4e+31)
		tmp = t_1;
	elseif (y <= 8.6e+84)
		tmp = t_0;
	else
		tmp = (y * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -1.1e-8], t$95$1, If[LessEqual[y, 50000.0], t$95$0, If[LessEqual[y, 3.4e+31], t$95$1, If[LessEqual[y, 8.6e+84], t$95$0, N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0999999999999999e-8 or 5e4 < y < 3.3999999999999998e31

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0 85.6%

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

   if -1.0999999999999999e-8 < y < 5e4 or 3.3999999999999998e31 < y < 8.5999999999999992e84

   1. Initial program 92.7%

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

   2. Taylor expanded in y around 0 90.4%

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

   3. Taylor expanded in z around 0 97.7%

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

      1. +-commutative 97.7%

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

      2. mul-1-neg 97.7%

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

      3. unsub-neg 97.7%

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

   5. Simplified 97.7%

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

   if 8.5999999999999992e84 < y

   1. Initial program 91.7%

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

   2. Taylor expanded in y around inf 83.4%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{elif}\;y \leq 50000:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+31}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternative 5: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-17} \lor \neg \left(z \leq 10^{-5}\right):\\ \;\;\;\;\frac{\left(y - z\right) + 1}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1e-17) (not (<= z 1e-5)))
   (* (/ (+ (- y z) 1.0) z) x)
   (/ (* x (+ y 1.0)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-17) || !(z <= 1e-5)) {
		tmp = (((y - z) + 1.0) / 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 <= (-1d-17)) .or. (.not. (z <= 1d-5))) then
        tmp = (((y - z) + 1.0d0) / 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 <= -1e-17) || !(z <= 1e-5)) {
		tmp = (((y - z) + 1.0) / z) * x;
	} else {
		tmp = (x * (y + 1.0)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1e-17) or not (z <= 1e-5):
		tmp = (((y - z) + 1.0) / z) * x
	else:
		tmp = (x * (y + 1.0)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1e-17) || !(z <= 1e-5))
		tmp = Float64(Float64(Float64(Float64(y - z) + 1.0) / 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 <= -1e-17) || ~((z <= 1e-5)))
		tmp = (((y - z) + 1.0) / z) * x;
	else
		tmp = (x * (y + 1.0)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1e-17], N[Not[LessEqual[z, 1e-5]], $MachinePrecision]], N[(N[(N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.00000000000000007e-17 or 1.00000000000000008e-5 < z

   1. Initial program 87.7%

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

      1. *-commutative 87.7%

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

      2. associate-/l* 71.9%

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

      3. associate-/r/ 99.8%

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

   3. Applied egg-rr 99.8%

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

   if -1.00000000000000007e-17 < z < 1.00000000000000008e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-17} \lor \neg \left(z \leq 10^{-5}\right):\\ \;\;\;\;\frac{\left(y - z\right) + 1}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \end{array} \]

Alternative 6: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;z \leq -2 \cdot 10^{+108} \lor \neg \left(z \leq 2.9 \cdot 10^{+64}\right):\\ \;\;\;\;\frac{t_0}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- y z) 1.0)))
   (if (or (<= z -2e+108) (not (<= z 2.9e+64)))
     (* (/ t_0 z) x)
     (/ (* t_0 x) z))))

C

double code(double x, double y, double z) {
	double t_0 = (y - z) + 1.0;
	double tmp;
	if ((z <= -2e+108) || !(z <= 2.9e+64)) {
		tmp = (t_0 / z) * x;
	} else {
		tmp = (t_0 * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y - z) + 1.0d0
    if ((z <= (-2d+108)) .or. (.not. (z <= 2.9d+64))) then
        tmp = (t_0 / z) * x
    else
        tmp = (t_0 * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - z) + 1.0;
	double tmp;
	if ((z <= -2e+108) || !(z <= 2.9e+64)) {
		tmp = (t_0 / z) * x;
	} else {
		tmp = (t_0 * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - z) + 1.0
	tmp = 0
	if (z <= -2e+108) or not (z <= 2.9e+64):
		tmp = (t_0 / z) * x
	else:
		tmp = (t_0 * x) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - z) + 1.0)
	tmp = 0.0
	if ((z <= -2e+108) || !(z <= 2.9e+64))
		tmp = Float64(Float64(t_0 / z) * x);
	else
		tmp = Float64(Float64(t_0 * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - z) + 1.0;
	tmp = 0.0;
	if ((z <= -2e+108) || ~((z <= 2.9e+64)))
		tmp = (t_0 / z) * x;
	else
		tmp = (t_0 * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]}, If[Or[LessEqual[z, -2e+108], N[Not[LessEqual[z, 2.9e+64]], $MachinePrecision]], N[(N[(t$95$0 / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(t$95$0 * x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0000000000000001e108 or 2.89999999999999993e64 < z

   1. Initial program 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-/l* 58.1%

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

      3. associate-/r/ 100.0%

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

   3. Applied egg-rr 100.0%

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

   if -2.0000000000000001e108 < z < 2.89999999999999993e64

   1. Initial program 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+108} \lor \neg \left(z \leq 2.9 \cdot 10^{+64}\right):\\ \;\;\;\;\frac{\left(y - z\right) + 1}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(y - z\right) + 1\right) \cdot x}{z}\\ \end{array} \]

Alternative 7: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -11500000 \lor \neg \left(y \leq 8.6 \cdot 10^{+84}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -11500000.0) (not (<= y 8.6e+84))) (* y (/ x z)) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -11500000.0) || !(y <= 8.6e+84)) {
		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 <= (-11500000.0d0)) .or. (.not. (y <= 8.6d+84))) 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 <= -11500000.0) || !(y <= 8.6e+84)) {
		tmp = y * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -11500000.0) or not (y <= 8.6e+84):
		tmp = y * (x / z)
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -11500000.0) || !(y <= 8.6e+84))
		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 <= -11500000.0) || ~((y <= 8.6e+84)))
		tmp = y * (x / z);
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -11500000.0], N[Not[LessEqual[y, 8.6e+84]], $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 < -1.15e7 or 8.5999999999999992e84 < y

   1. Initial program 94.0%

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

   2. Taylor expanded in y around inf 83.9%

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

      1. associate-*l/ 80.7%

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

      2. *-commutative 80.7%

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

   4. Simplified 80.7%

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

   if -1.15e7 < y < 8.5999999999999992e84

   1. Initial program 93.3%

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

   2. Taylor expanded in y around 0 86.0%

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

   3. Taylor expanded in z around 0 92.6%

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

      1. +-commutative 92.6%

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

      2. mul-1-neg 92.6%

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

      3. unsub-neg 92.6%

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

   5. Simplified 92.6%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11500000 \lor \neg \left(y \leq 8.6 \cdot 10^{+84}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 8: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1300000 \lor \neg \left(y \leq 8.6 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1300000.0) (not (<= y 8.6e+84))) (/ (* y x) z) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1300000.0) || !(y <= 8.6e+84)) {
		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 <= (-1300000.0d0)) .or. (.not. (y <= 8.6d+84))) 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 <= -1300000.0) || !(y <= 8.6e+84)) {
		tmp = (y * x) / z;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1300000.0) or not (y <= 8.6e+84):
		tmp = (y * x) / z
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1300000.0) || !(y <= 8.6e+84))
		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 <= -1300000.0) || ~((y <= 8.6e+84)))
		tmp = (y * x) / z;
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1300000.0], N[Not[LessEqual[y, 8.6e+84]], $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 < -1.3e6 or 8.5999999999999992e84 < y

   1. Initial program 94.0%

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

   2. Taylor expanded in y around inf 83.9%

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

   if -1.3e6 < y < 8.5999999999999992e84

   1. Initial program 93.3%

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

   2. Taylor expanded in y around 0 86.0%

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

   3. Taylor expanded in z around 0 92.6%

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

      1. +-commutative 92.6%

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

      2. mul-1-neg 92.6%

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

      3. unsub-neg 92.6%

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

   5. Simplified 92.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1300000 \lor \neg \left(y \leq 8.6 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 9: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5000000:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5000000.0)
   (* y (/ x z))
   (if (<= y 9e+84) (- (/ x z) x) (/ y (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5000000.0) {
		tmp = y * (x / z);
	} else if (y <= 9e+84) {
		tmp = (x / z) - x;
	} else {
		tmp = y / (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 <= (-5000000.0d0)) then
        tmp = y * (x / z)
    else if (y <= 9d+84) then
        tmp = (x / z) - x
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5000000.0) {
		tmp = y * (x / z);
	} else if (y <= 9e+84) {
		tmp = (x / z) - x;
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5000000.0:
		tmp = y * (x / z)
	elif y <= 9e+84:
		tmp = (x / z) - x
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5000000.0)
		tmp = Float64(y * Float64(x / z));
	elseif (y <= 9e+84)
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5000000.0)
		tmp = y * (x / z);
	elseif (y <= 9e+84)
		tmp = (x / z) - x;
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5000000.0], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e+84], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5e6

   1. Initial program 96.2%

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

   2. Taylor expanded in y around inf 84.4%

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

      1. associate-*l/ 78.6%

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

      2. *-commutative 78.6%

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

   4. Simplified 78.6%

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

   if -5e6 < y < 8.9999999999999994e84

   1. Initial program 93.3%

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

   2. Taylor expanded in y around 0 86.0%

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

   3. Taylor expanded in z around 0 92.6%

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

      1. +-commutative 92.6%

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

      2. mul-1-neg 92.6%

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

      3. unsub-neg 92.6%

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

   5. Simplified 92.6%

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

   if 8.9999999999999994e84 < y

   1. Initial program 91.7%

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

      1. *-commutative 91.7%

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

      2. associate-/l* 87.3%

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

      3. associate-/r/ 89.6%

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

   3. Applied egg-rr 89.6%

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

   4. Taylor expanded in y around inf 79.3%

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

      1. associate-/r/ 83.3%

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

   6. Applied egg-rr 83.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5000000:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 10: 65.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 9500\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 9500.0))) (- x) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 9500.0)) {
		tmp = -x;
	} else {
		tmp = 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 <= (-1.0d0)) .or. (.not. (z <= 9500.0d0))) then
        tmp = -x
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 9500.0)) {
		tmp = -x;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 9500.0):
		tmp = -x
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 9500.0))
		tmp = Float64(-x);
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 9500.0)))
		tmp = -x;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 9500.0]], $MachinePrecision]], (-x), N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 9500 < z

   1. Initial program 87.2%

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

   2. Taylor expanded in z around inf 68.6%

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

      1. mul-1-neg 68.6%

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

   4. Simplified 68.6%

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

   if -1 < z < 9500

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 98.3%

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

   3. Taylor expanded in y around 0 57.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 9500\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 11: 38.7% 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 93.6%

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

2. Taylor expanded in z around inf 35.7%

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

   1. mul-1-neg 35.7%

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

4. Simplified 35.7%

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

5. Final simplification 35.7%

   \[\leadsto -x \]

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 2023301 
(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))