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

Percentage Accurate: 87.6% → 99.8%

Time: 9.7s

Alternatives: 9

Speedup: 0.7×

• 20.5% 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: 87.6% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0002 \lor \neg \left(z \leq 4.6 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.0002) (not (<= z 4.6e+16)))
   (/ x (/ z (+ 1.0 (- y z))))
   (/ (fma x (- y z) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.0002) || !(z <= 4.6e+16)) {
		tmp = x / (z / (1.0 + (y - z)));
	} else {
		tmp = fma(x, (y - z), x) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.0002) || !(z <= 4.6e+16))
		tmp = Float64(x / Float64(z / Float64(1.0 + Float64(y - z))));
	else
		tmp = Float64(fma(x, Float64(y - z), x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.0002], N[Not[LessEqual[z, 4.6e+16]], $MachinePrecision]], N[(x / N[(z / N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y - z), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0000000000000001e-4 or 4.6e16 < z

   1. Initial program 78.3%

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

      1. div-inv 78.2%

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

      2. associate-*l* 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 78.3%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

   6. Simplified 100.0%

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

   if -2.0000000000000001e-4 < z < 4.6e16

   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 99.9%

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

      2. fma-def 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0002 \lor \neg \left(z \leq 4.6 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \end{array} \]

Alternative 2: 64.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+31}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -3.05 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ y z))))
   (if (<= z -6.8e+31)
     (- x)
     (if (<= z -6e-61)
       t_0
       (if (<= z -2.6e-150)
         (/ x z)
         (if (<= z -3.05e-192)
           t_0
           (if (<= z 2.4e-178)
             (/ x z)
             (if (<= z 6.1e-32)
               t_0
               (if (<= z 3.3e-10)
                 (/ x z)
                 (if (<= z 4.2e+62) t_0 (- x)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y / z);
	double tmp;
	if (z <= -6.8e+31) {
		tmp = -x;
	} else if (z <= -6e-61) {
		tmp = t_0;
	} else if (z <= -2.6e-150) {
		tmp = x / z;
	} else if (z <= -3.05e-192) {
		tmp = t_0;
	} else if (z <= 2.4e-178) {
		tmp = x / z;
	} else if (z <= 6.1e-32) {
		tmp = t_0;
	} else if (z <= 3.3e-10) {
		tmp = x / z;
	} else if (z <= 4.2e+62) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y / z)
    if (z <= (-6.8d+31)) then
        tmp = -x
    else if (z <= (-6d-61)) then
        tmp = t_0
    else if (z <= (-2.6d-150)) then
        tmp = x / z
    else if (z <= (-3.05d-192)) then
        tmp = t_0
    else if (z <= 2.4d-178) then
        tmp = x / z
    else if (z <= 6.1d-32) then
        tmp = t_0
    else if (z <= 3.3d-10) then
        tmp = x / z
    else if (z <= 4.2d+62) then
        tmp = t_0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y / z);
	double tmp;
	if (z <= -6.8e+31) {
		tmp = -x;
	} else if (z <= -6e-61) {
		tmp = t_0;
	} else if (z <= -2.6e-150) {
		tmp = x / z;
	} else if (z <= -3.05e-192) {
		tmp = t_0;
	} else if (z <= 2.4e-178) {
		tmp = x / z;
	} else if (z <= 6.1e-32) {
		tmp = t_0;
	} else if (z <= 3.3e-10) {
		tmp = x / z;
	} else if (z <= 4.2e+62) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y / z)
	tmp = 0
	if z <= -6.8e+31:
		tmp = -x
	elif z <= -6e-61:
		tmp = t_0
	elif z <= -2.6e-150:
		tmp = x / z
	elif z <= -3.05e-192:
		tmp = t_0
	elif z <= 2.4e-178:
		tmp = x / z
	elif z <= 6.1e-32:
		tmp = t_0
	elif z <= 3.3e-10:
		tmp = x / z
	elif z <= 4.2e+62:
		tmp = t_0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y / z))
	tmp = 0.0
	if (z <= -6.8e+31)
		tmp = Float64(-x);
	elseif (z <= -6e-61)
		tmp = t_0;
	elseif (z <= -2.6e-150)
		tmp = Float64(x / z);
	elseif (z <= -3.05e-192)
		tmp = t_0;
	elseif (z <= 2.4e-178)
		tmp = Float64(x / z);
	elseif (z <= 6.1e-32)
		tmp = t_0;
	elseif (z <= 3.3e-10)
		tmp = Float64(x / z);
	elseif (z <= 4.2e+62)
		tmp = t_0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y / z);
	tmp = 0.0;
	if (z <= -6.8e+31)
		tmp = -x;
	elseif (z <= -6e-61)
		tmp = t_0;
	elseif (z <= -2.6e-150)
		tmp = x / z;
	elseif (z <= -3.05e-192)
		tmp = t_0;
	elseif (z <= 2.4e-178)
		tmp = x / z;
	elseif (z <= 6.1e-32)
		tmp = t_0;
	elseif (z <= 3.3e-10)
		tmp = x / z;
	elseif (z <= 4.2e+62)
		tmp = t_0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.8e+31], (-x), If[LessEqual[z, -6e-61], t$95$0, If[LessEqual[z, -2.6e-150], N[(x / z), $MachinePrecision], If[LessEqual[z, -3.05e-192], t$95$0, If[LessEqual[z, 2.4e-178], N[(x / z), $MachinePrecision], If[LessEqual[z, 6.1e-32], t$95$0, If[LessEqual[z, 3.3e-10], N[(x / z), $MachinePrecision], If[LessEqual[z, 4.2e+62], t$95$0, (-x)]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.7999999999999996e31 or 4.2e62 < z

   1. Initial program 74.5%

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

   2. Taylor expanded in z around inf 77.5%

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

      1. neg-mul-1 77.5%

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

   4. Simplified 77.5%

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

   if -6.7999999999999996e31 < z < -6.00000000000000024e-61 or -2.5999999999999998e-150 < z < -3.05e-192 or 2.40000000000000005e-178 < z < 6.09999999999999959e-32 or 3.3e-10 < z < 4.2e62

   1. Initial program 98.6%

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

      1. div-inv 98.5%

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

      2. associate-*l* 93.7%

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

   3. Applied egg-rr 93.7%

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

   4. Taylor expanded in y around inf 73.2%

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

      1. associate-*r/ 68.4%

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

   6. Simplified 68.4%

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

   if -6.00000000000000024e-61 < z < -2.5999999999999998e-150 or -3.05e-192 < z < 2.40000000000000005e-178 or 6.09999999999999959e-32 < z < 3.3e-10

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 71.2%

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

      1. associate-/l* 71.2%

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

      2. associate-/r/ 71.2%

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

   4. Simplified 71.2%

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

   5. Taylor expanded in z around 0 70.7%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+31}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -3.05 \cdot 10^{-192}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 3: 65.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -4.3 \cdot 10^{+26}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-175}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+63}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))))
   (if (<= z -4.3e+26)
     (- x)
     (if (<= z -2.3e-68)
       t_0
       (if (<= z -5e-147)
         (/ x z)
         (if (<= z 3.7e-296)
           t_0
           (if (<= z 5e-175)
             (/ x z)
             (if (<= z 6e-36)
               t_0
               (if (<= z 1.9e-9)
                 (/ x z)
                 (if (<= z 5.9e+63) (* x (/ y z)) (- x)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (z <= -4.3e+26) {
		tmp = -x;
	} else if (z <= -2.3e-68) {
		tmp = t_0;
	} else if (z <= -5e-147) {
		tmp = x / z;
	} else if (z <= 3.7e-296) {
		tmp = t_0;
	} else if (z <= 5e-175) {
		tmp = x / z;
	} else if (z <= 6e-36) {
		tmp = t_0;
	} else if (z <= 1.9e-9) {
		tmp = x / z;
	} else if (z <= 5.9e+63) {
		tmp = x * (y / 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 <= (-4.3d+26)) then
        tmp = -x
    else if (z <= (-2.3d-68)) then
        tmp = t_0
    else if (z <= (-5d-147)) then
        tmp = x / z
    else if (z <= 3.7d-296) then
        tmp = t_0
    else if (z <= 5d-175) then
        tmp = x / z
    else if (z <= 6d-36) then
        tmp = t_0
    else if (z <= 1.9d-9) then
        tmp = x / z
    else if (z <= 5.9d+63) then
        tmp = x * (y / 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 <= -4.3e+26) {
		tmp = -x;
	} else if (z <= -2.3e-68) {
		tmp = t_0;
	} else if (z <= -5e-147) {
		tmp = x / z;
	} else if (z <= 3.7e-296) {
		tmp = t_0;
	} else if (z <= 5e-175) {
		tmp = x / z;
	} else if (z <= 6e-36) {
		tmp = t_0;
	} else if (z <= 1.9e-9) {
		tmp = x / z;
	} else if (z <= 5.9e+63) {
		tmp = x * (y / z);
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / z)
	tmp = 0
	if z <= -4.3e+26:
		tmp = -x
	elif z <= -2.3e-68:
		tmp = t_0
	elif z <= -5e-147:
		tmp = x / z
	elif z <= 3.7e-296:
		tmp = t_0
	elif z <= 5e-175:
		tmp = x / z
	elif z <= 6e-36:
		tmp = t_0
	elif z <= 1.9e-9:
		tmp = x / z
	elif z <= 5.9e+63:
		tmp = x * (y / z)
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (z <= -4.3e+26)
		tmp = Float64(-x);
	elseif (z <= -2.3e-68)
		tmp = t_0;
	elseif (z <= -5e-147)
		tmp = Float64(x / z);
	elseif (z <= 3.7e-296)
		tmp = t_0;
	elseif (z <= 5e-175)
		tmp = Float64(x / z);
	elseif (z <= 6e-36)
		tmp = t_0;
	elseif (z <= 1.9e-9)
		tmp = Float64(x / z);
	elseif (z <= 5.9e+63)
		tmp = Float64(x * Float64(y / 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 <= -4.3e+26)
		tmp = -x;
	elseif (z <= -2.3e-68)
		tmp = t_0;
	elseif (z <= -5e-147)
		tmp = x / z;
	elseif (z <= 3.7e-296)
		tmp = t_0;
	elseif (z <= 5e-175)
		tmp = x / z;
	elseif (z <= 6e-36)
		tmp = t_0;
	elseif (z <= 1.9e-9)
		tmp = x / z;
	elseif (z <= 5.9e+63)
		tmp = x * (y / 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, -4.3e+26], (-x), If[LessEqual[z, -2.3e-68], t$95$0, If[LessEqual[z, -5e-147], N[(x / z), $MachinePrecision], If[LessEqual[z, 3.7e-296], t$95$0, If[LessEqual[z, 5e-175], N[(x / z), $MachinePrecision], If[LessEqual[z, 6e-36], t$95$0, If[LessEqual[z, 1.9e-9], N[(x / z), $MachinePrecision], If[LessEqual[z, 5.9e+63], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], (-x)]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.2999999999999998e26 or 5.90000000000000029e63 < z

   1. Initial program 75.0%

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

   2. Taylor expanded in z around inf 77.0%

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

      1. neg-mul-1 77.0%

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

   4. Simplified 77.0%

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

   if -4.2999999999999998e26 < z < -2.29999999999999997e-68 or -5.00000000000000013e-147 < z < 3.70000000000000027e-296 or 5e-175 < z < 6.0000000000000003e-36

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 73.4%

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

      1. associate-/l* 64.0%

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

      2. associate-/r/ 74.7%

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

   4. Simplified 74.7%

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

   if -2.29999999999999997e-68 < z < -5.00000000000000013e-147 or 3.70000000000000027e-296 < z < 5e-175 or 6.0000000000000003e-36 < z < 1.90000000000000006e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 78.2%

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

      1. associate-/l* 78.2%

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

      2. associate-/r/ 78.2%

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

   4. Simplified 78.2%

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

   5. Taylor expanded in z around 0 77.5%

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

   if 1.90000000000000006e-9 < z < 5.90000000000000029e63

   1. Initial program 94.5%

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

      1. div-inv 94.5%

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

      2. associate-*l* 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in y around inf 62.8%

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

      1. associate-*r/ 68.0%

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

   6. Simplified 68.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+26}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-68}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-296}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-175}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-36}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+63}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y - z\right)\\ \mathbf{if}\;z \leq -6 \cdot 10^{+27} \lor \neg \left(z \leq 1.8 \cdot 10^{-111}\right):\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (- y z))))
   (if (or (<= z -6e+27) (not (<= z 1.8e-111)))
     (/ x (/ z t_0))
     (* t_0 (/ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (y - z);
	double tmp;
	if ((z <= -6e+27) || !(z <= 1.8e-111)) {
		tmp = x / (z / t_0);
	} else {
		tmp = t_0 * (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (y - z)
    if ((z <= (-6d+27)) .or. (.not. (z <= 1.8d-111))) then
        tmp = x / (z / t_0)
    else
        tmp = t_0 * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (y - z);
	double tmp;
	if ((z <= -6e+27) || !(z <= 1.8e-111)) {
		tmp = x / (z / t_0);
	} else {
		tmp = t_0 * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (y - z)
	tmp = 0
	if (z <= -6e+27) or not (z <= 1.8e-111):
		tmp = x / (z / t_0)
	else:
		tmp = t_0 * (x / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(y - z))
	tmp = 0.0
	if ((z <= -6e+27) || !(z <= 1.8e-111))
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(t_0 * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (y - z);
	tmp = 0.0;
	if ((z <= -6e+27) || ~((z <= 1.8e-111)))
		tmp = x / (z / t_0);
	else
		tmp = t_0 * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[z, -6e+27], N[Not[LessEqual[z, 1.8e-111]], $MachinePrecision]], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.99999999999999953e27 or 1.80000000000000005e-111 < z

   1. Initial program 80.4%

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

      1. div-inv 80.3%

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

      2. associate-*l* 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 80.4%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

   6. Simplified 100.0%

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

   if -5.99999999999999953e27 < z < 1.80000000000000005e-111

   1. Initial program 99.8%

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

      1. associate-/l* 90.1%

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

      2. associate-/r/ 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+27} \lor \neg \left(z \leq 1.8 \cdot 10^{-111}\right):\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(y - z\right)\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 5: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+182}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+146}:\\ \;\;\;\;\left(1 + \left(y - z\right)\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.06e+182)
   (- x)
   (if (<= z 7e+146) (* (+ 1.0 (- y z)) (/ x z)) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.06e+182) {
		tmp = -x;
	} else if (z <= 7e+146) {
		tmp = (1.0 + (y - z)) * (x / z);
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.06d+182)) then
        tmp = -x
    else if (z <= 7d+146) then
        tmp = (1.0d0 + (y - z)) * (x / z)
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.06e+182) {
		tmp = -x;
	} else if (z <= 7e+146) {
		tmp = (1.0 + (y - z)) * (x / z);
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.06e+182:
		tmp = -x
	elif z <= 7e+146:
		tmp = (1.0 + (y - z)) * (x / z)
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.06e+182)
		tmp = Float64(-x);
	elseif (z <= 7e+146)
		tmp = Float64(Float64(1.0 + Float64(y - z)) * Float64(x / z));
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.06e+182)
		tmp = -x;
	elseif (z <= 7e+146)
		tmp = (1.0 + (y - z)) * (x / z);
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.06e+182], (-x), If[LessEqual[z, 7e+146], N[(N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], (-x)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0599999999999999e182 or 7.0000000000000002e146 < z

   1. Initial program 72.1%

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

   2. Taylor expanded in z around inf 84.3%

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

      1. neg-mul-1 84.3%

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

   4. Simplified 84.3%

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

   if -1.0599999999999999e182 < z < 7.0000000000000002e146

   1. Initial program 93.7%

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

      1. associate-/l* 94.4%

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

      2. associate-/r/ 96.3%

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

   3. Applied egg-rr 96.3%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+182}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+146}:\\ \;\;\;\;\left(1 + \left(y - z\right)\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 6: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+38} \lor \neg \left(y \leq 1.14 \cdot 10^{+24}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.2e+38) (not (<= y 1.14e+24))) (* y (/ x z)) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.2e+38) || !(y <= 1.14e+24)) {
		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 <= (-6.2d+38)) .or. (.not. (y <= 1.14d+24))) 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 <= -6.2e+38) || !(y <= 1.14e+24)) {
		tmp = y * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.2e+38) or not (y <= 1.14e+24):
		tmp = y * (x / z)
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.2e+38) || !(y <= 1.14e+24))
		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 <= -6.2e+38) || ~((y <= 1.14e+24)))
		tmp = y * (x / z);
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.2e+38], N[Not[LessEqual[y, 1.14e+24]], $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 < -6.20000000000000035e38 or 1.14e24 < y

   1. Initial program 91.4%

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

   2. Taylor expanded in y around inf 80.6%

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

      1. associate-/l* 75.8%

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

      2. associate-/r/ 80.0%

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

   4. Simplified 80.0%

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

   if -6.20000000000000035e38 < y < 1.14e24

   1. Initial program 86.5%

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

      1. div-inv 86.3%

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

      2. associate-*l* 99.0%

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

   3. Applied egg-rr 99.0%

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

   4. Taylor expanded in y around 0 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-/l* 81.8%

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

      3. div-sub 74.4%

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

      4. associate-/r/ 74.3%

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

      5. associate-*l/ 74.5%

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

      6. *-lft-identity 74.5%

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

      7. associate-/r/ 95.3%

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

      8. *-inverses 95.3%

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

      9. *-lft-identity 95.3%

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

   6. Simplified 95.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+38} \lor \neg \left(y \leq 1.14 \cdot 10^{+24}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 7: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+41} \lor \neg \left(y \leq 1.1 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.4e+41) (not (<= y 1.1e+24))) (/ (* x y) z) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.4e+41) || !(y <= 1.1e+24)) {
		tmp = (x * y) / 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 <= (-1.4d+41)) .or. (.not. (y <= 1.1d+24))) then
        tmp = (x * y) / 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 <= -1.4e+41) || !(y <= 1.1e+24)) {
		tmp = (x * y) / z;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.4e+41) or not (y <= 1.1e+24):
		tmp = (x * y) / z
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.4e+41) || !(y <= 1.1e+24))
		tmp = Float64(Float64(x * y) / 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 <= -1.4e+41) || ~((y <= 1.1e+24)))
		tmp = (x * y) / z;
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.4e+41], N[Not[LessEqual[y, 1.1e+24]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e41 or 1.10000000000000001e24 < y

   1. Initial program 91.4%

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

   2. Taylor expanded in y around inf 80.6%

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

   if -1.4e41 < y < 1.10000000000000001e24

   1. Initial program 86.5%

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

      1. div-inv 86.3%

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

      2. associate-*l* 99.0%

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

   3. Applied egg-rr 99.0%

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

   4. Taylor expanded in y around 0 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-/l* 81.8%

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

      3. div-sub 74.4%

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

      4. associate-/r/ 74.3%

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

      5. associate-*l/ 74.5%

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

      6. *-lft-identity 74.5%

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

      7. associate-/r/ 95.3%

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

      8. *-inverses 95.3%

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

      9. *-lft-identity 95.3%

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

   6. Simplified 95.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+41} \lor \neg \left(y \leq 1.1 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 8: 64.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 78.6%

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

   2. Taylor expanded in z around inf 68.4%

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

      1. neg-mul-1 68.4%

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

   4. Simplified 68.4%

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

   if -1 < z < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 56.3%

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

      1. associate-/l* 56.3%

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

      2. associate-/r/ 56.3%

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

   4. Simplified 56.3%

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

   5. Taylor expanded in z around 0 55.1%

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

4. Final simplification 62.0%

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

Alternative 9: 39.4% 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 88.8%

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

2. Taylor expanded in z around inf 37.2%

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

   1. neg-mul-1 37.2%

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

4. Simplified 37.2%

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

5. Final simplification 37.2%

   \[\leadsto -x \]

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 2023280 
(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))