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

Percentage Accurate: 87.6% → 99.9%

Time: 8.2s

Alternatives: 14

Speedup: 0.6×

• 20.6% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-17} \lor \neg \left(z \leq 9.2 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{x}{\frac{z}{1 + y}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.5e-17) (not (<= z 9.2e-48)))
   (- (/ x (/ z (+ 1.0 y))) x)
   (/ (+ x (* x y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.5e-17) || !(z <= 9.2e-48)) {
		tmp = (x / (z / (1.0 + y))) - x;
	} else {
		tmp = (x + (x * y)) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.5d-17)) .or. (.not. (z <= 9.2d-48))) then
        tmp = (x / (z / (1.0d0 + y))) - x
    else
        tmp = (x + (x * y)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.5e-17) || !(z <= 9.2e-48)) {
		tmp = (x / (z / (1.0 + y))) - x;
	} else {
		tmp = (x + (x * y)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.5e-17) or not (z <= 9.2e-48):
		tmp = (x / (z / (1.0 + y))) - x
	else:
		tmp = (x + (x * y)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.5e-17) || !(z <= 9.2e-48))
		tmp = Float64(Float64(x / Float64(z / Float64(1.0 + y))) - x);
	else
		tmp = Float64(Float64(x + Float64(x * y)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.5e-17) || ~((z <= 9.2e-48)))
		tmp = (x / (z / (1.0 + y))) - x;
	else
		tmp = (x + (x * y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.5e-17], N[Not[LessEqual[z, 9.2e-48]], $MachinePrecision]], N[(N[(x / N[(z / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.50000000000000003e-17 or 9.2000000000000003e-48 < z

   1. Initial program 75.5%

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

   3. Taylor expanded in x around 0 75.5%

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

      1. associate--l+ 75.5%

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

      2. +-commutative 75.5%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--l+ 99.8%

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

      6. div-sub 99.8%

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

      7. sub-neg 99.8%

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

      8. *-inverses 99.8%

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

      9. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. neg-mul-1 99.8%

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

      3. unsub-neg 99.8%

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

      4. clear-num 99.8%

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

      5. associate-*l/ 99.9%

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

      6. *-un-lft-identity 99.9%

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

   7. Applied egg-rr 99.9%

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

   if -1.50000000000000003e-17 < z < 9.2000000000000003e-48

   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%

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

   5. Taylor expanded in z around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-17} \lor \neg \left(z \leq 9.2 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{x}{\frac{z}{1 + y}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ t_1 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+77}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2.55 \cdot 10^{-79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-267}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-229}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-208}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{1}{z}\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))) (t_1 (* x (/ y z))))
   (if (<= z -3.2e+77)
     (- x)
     (if (<= z -2.55e-79)
       t_1
       (if (<= z -2.5e-93)
         (/ x z)
         (if (<= z -1.65e-267)
           t_0
           (if (<= z 1.55e-229)
             (/ x z)
             (if (<= z 1.4e-208)
               t_0
               (if (<= z 3.25e-159)
                 (* x (/ 1.0 z))
                 (if (<= z 6.3e-65)
                   t_0
                   (if (<= z 8.6e-13)
                     (/ x z)
                     (if (<= z 1.35e+105) t_1 (- x)))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double t_1 = x * (y / z);
	double tmp;
	if (z <= -3.2e+77) {
		tmp = -x;
	} else if (z <= -2.55e-79) {
		tmp = t_1;
	} else if (z <= -2.5e-93) {
		tmp = x / z;
	} else if (z <= -1.65e-267) {
		tmp = t_0;
	} else if (z <= 1.55e-229) {
		tmp = x / z;
	} else if (z <= 1.4e-208) {
		tmp = t_0;
	} else if (z <= 3.25e-159) {
		tmp = x * (1.0 / z);
	} else if (z <= 6.3e-65) {
		tmp = t_0;
	} else if (z <= 8.6e-13) {
		tmp = x / z;
	} else if (z <= 1.35e+105) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = y * (x / z)
    t_1 = x * (y / z)
    if (z <= (-3.2d+77)) then
        tmp = -x
    else if (z <= (-2.55d-79)) then
        tmp = t_1
    else if (z <= (-2.5d-93)) then
        tmp = x / z
    else if (z <= (-1.65d-267)) then
        tmp = t_0
    else if (z <= 1.55d-229) then
        tmp = x / z
    else if (z <= 1.4d-208) then
        tmp = t_0
    else if (z <= 3.25d-159) then
        tmp = x * (1.0d0 / z)
    else if (z <= 6.3d-65) then
        tmp = t_0
    else if (z <= 8.6d-13) then
        tmp = x / z
    else if (z <= 1.35d+105) then
        tmp = t_1
    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 t_1 = x * (y / z);
	double tmp;
	if (z <= -3.2e+77) {
		tmp = -x;
	} else if (z <= -2.55e-79) {
		tmp = t_1;
	} else if (z <= -2.5e-93) {
		tmp = x / z;
	} else if (z <= -1.65e-267) {
		tmp = t_0;
	} else if (z <= 1.55e-229) {
		tmp = x / z;
	} else if (z <= 1.4e-208) {
		tmp = t_0;
	} else if (z <= 3.25e-159) {
		tmp = x * (1.0 / z);
	} else if (z <= 6.3e-65) {
		tmp = t_0;
	} else if (z <= 8.6e-13) {
		tmp = x / z;
	} else if (z <= 1.35e+105) {
		tmp = t_1;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / z)
	t_1 = x * (y / z)
	tmp = 0
	if z <= -3.2e+77:
		tmp = -x
	elif z <= -2.55e-79:
		tmp = t_1
	elif z <= -2.5e-93:
		tmp = x / z
	elif z <= -1.65e-267:
		tmp = t_0
	elif z <= 1.55e-229:
		tmp = x / z
	elif z <= 1.4e-208:
		tmp = t_0
	elif z <= 3.25e-159:
		tmp = x * (1.0 / z)
	elif z <= 6.3e-65:
		tmp = t_0
	elif z <= 8.6e-13:
		tmp = x / z
	elif z <= 1.35e+105:
		tmp = t_1
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	t_1 = Float64(x * Float64(y / z))
	tmp = 0.0
	if (z <= -3.2e+77)
		tmp = Float64(-x);
	elseif (z <= -2.55e-79)
		tmp = t_1;
	elseif (z <= -2.5e-93)
		tmp = Float64(x / z);
	elseif (z <= -1.65e-267)
		tmp = t_0;
	elseif (z <= 1.55e-229)
		tmp = Float64(x / z);
	elseif (z <= 1.4e-208)
		tmp = t_0;
	elseif (z <= 3.25e-159)
		tmp = Float64(x * Float64(1.0 / z));
	elseif (z <= 6.3e-65)
		tmp = t_0;
	elseif (z <= 8.6e-13)
		tmp = Float64(x / z);
	elseif (z <= 1.35e+105)
		tmp = t_1;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x / z);
	t_1 = x * (y / z);
	tmp = 0.0;
	if (z <= -3.2e+77)
		tmp = -x;
	elseif (z <= -2.55e-79)
		tmp = t_1;
	elseif (z <= -2.5e-93)
		tmp = x / z;
	elseif (z <= -1.65e-267)
		tmp = t_0;
	elseif (z <= 1.55e-229)
		tmp = x / z;
	elseif (z <= 1.4e-208)
		tmp = t_0;
	elseif (z <= 3.25e-159)
		tmp = x * (1.0 / z);
	elseif (z <= 6.3e-65)
		tmp = t_0;
	elseif (z <= 8.6e-13)
		tmp = x / z;
	elseif (z <= 1.35e+105)
		tmp = t_1;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.2e+77], (-x), If[LessEqual[z, -2.55e-79], t$95$1, If[LessEqual[z, -2.5e-93], N[(x / z), $MachinePrecision], If[LessEqual[z, -1.65e-267], t$95$0, If[LessEqual[z, 1.55e-229], N[(x / z), $MachinePrecision], If[LessEqual[z, 1.4e-208], t$95$0, If[LessEqual[z, 3.25e-159], N[(x * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.3e-65], t$95$0, If[LessEqual[z, 8.6e-13], N[(x / z), $MachinePrecision], If[LessEqual[z, 1.35e+105], t$95$1, (-x)]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -3.2000000000000002e77 or 1.35000000000000008e105 < z

   1. Initial program 65.6%

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

   3. Taylor expanded in z around inf 80.8%

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

      1. mul-1-neg 80.8%

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

   5. Simplified 80.8%

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

   if -3.2000000000000002e77 < z < -2.55e-79 or 8.5999999999999997e-13 < z < 1.35000000000000008e105

   1. Initial program 91.6%

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

   3. Taylor expanded in y around inf 62.9%

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

      1. associate-/l* 69.4%

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

   5. Simplified 69.4%

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

      1. clear-num 69.3%

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

      2. associate-/r/ 69.3%

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

      3. clear-num 69.3%

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

   7. Applied egg-rr 69.3%

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

   if -2.55e-79 < z < -2.49999999999999997e-93 or -1.65000000000000002e-267 < z < 1.55e-229 or 6.2999999999999997e-65 < z < 8.5999999999999997e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 87.0%

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

   4. Taylor expanded in z around 0 86.6%

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

   if -2.49999999999999997e-93 < z < -1.65000000000000002e-267 or 1.55e-229 < z < 1.40000000000000001e-208 or 3.2500000000000001e-159 < z < 6.2999999999999997e-65

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 70.9%

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

      1. associate-/l* 57.6%

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

      2. associate-/r/ 74.1%

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

   5. Simplified 74.1%

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

   if 1.40000000000000001e-208 < z < 3.2500000000000001e-159

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 93.5%

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

      1. *-un-lft-identity 93.5%

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

      2. times-frac 93.5%

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

      3. add-sqr-sqrt 33.0%

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

      4. sqrt-unprod 23.0%

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

      5. sqr-neg 23.0%

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

      6. sqrt-unprod 0.3%

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

      7. add-sqr-sqrt 0.6%

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

      8. div-inv 0.6%

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

      9. metadata-eval 0.6%

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

      10. *-commutative 0.6%

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

      11. *-un-lft-identity 0.6%

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

      12. add-sqr-sqrt 0.3%

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

      13. sqrt-unprod 23.0%

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

      14. sqr-neg 23.0%

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

      15. sqrt-unprod 33.0%

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

      16. add-sqr-sqrt 93.5%

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

   5. Applied egg-rr 93.5%

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

   6. Taylor expanded in z around 0 93.5%

      \[\leadsto x \cdot \color{blue}{\frac{1}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+77}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2.55 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-267}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-229}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-208}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{1}{z}\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{-65}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -5.6 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-130}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-162}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-199}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-267}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-266}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-78}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))))
   (if (<= y -5.6e+36)
     t_0
     (if (<= y -7.2e-130)
       (- x)
       (if (<= y -1.45e-162)
         (/ x z)
         (if (<= y -4.1e-199)
           (- x)
           (if (<= y -2.25e-267)
             (/ x z)
             (if (<= y 2.75e-266)
               (- x)
               (if (<= y 4.2e-95)
                 (/ x z)
                 (if (<= y 5.2e-78) (- x) (if (<= y 1.0) (/ x z) t_0)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (y <= -5.6e+36) {
		tmp = t_0;
	} else if (y <= -7.2e-130) {
		tmp = -x;
	} else if (y <= -1.45e-162) {
		tmp = x / z;
	} else if (y <= -4.1e-199) {
		tmp = -x;
	} else if (y <= -2.25e-267) {
		tmp = x / z;
	} else if (y <= 2.75e-266) {
		tmp = -x;
	} else if (y <= 4.2e-95) {
		tmp = x / z;
	} else if (y <= 5.2e-78) {
		tmp = -x;
	} else if (y <= 1.0) {
		tmp = x / 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 = y * (x / z)
    if (y <= (-5.6d+36)) then
        tmp = t_0
    else if (y <= (-7.2d-130)) then
        tmp = -x
    else if (y <= (-1.45d-162)) then
        tmp = x / z
    else if (y <= (-4.1d-199)) then
        tmp = -x
    else if (y <= (-2.25d-267)) then
        tmp = x / z
    else if (y <= 2.75d-266) then
        tmp = -x
    else if (y <= 4.2d-95) then
        tmp = x / z
    else if (y <= 5.2d-78) then
        tmp = -x
    else if (y <= 1.0d0) then
        tmp = x / 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 = y * (x / z);
	double tmp;
	if (y <= -5.6e+36) {
		tmp = t_0;
	} else if (y <= -7.2e-130) {
		tmp = -x;
	} else if (y <= -1.45e-162) {
		tmp = x / z;
	} else if (y <= -4.1e-199) {
		tmp = -x;
	} else if (y <= -2.25e-267) {
		tmp = x / z;
	} else if (y <= 2.75e-266) {
		tmp = -x;
	} else if (y <= 4.2e-95) {
		tmp = x / z;
	} else if (y <= 5.2e-78) {
		tmp = -x;
	} else if (y <= 1.0) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / z)
	tmp = 0
	if y <= -5.6e+36:
		tmp = t_0
	elif y <= -7.2e-130:
		tmp = -x
	elif y <= -1.45e-162:
		tmp = x / z
	elif y <= -4.1e-199:
		tmp = -x
	elif y <= -2.25e-267:
		tmp = x / z
	elif y <= 2.75e-266:
		tmp = -x
	elif y <= 4.2e-95:
		tmp = x / z
	elif y <= 5.2e-78:
		tmp = -x
	elif y <= 1.0:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (y <= -5.6e+36)
		tmp = t_0;
	elseif (y <= -7.2e-130)
		tmp = Float64(-x);
	elseif (y <= -1.45e-162)
		tmp = Float64(x / z);
	elseif (y <= -4.1e-199)
		tmp = Float64(-x);
	elseif (y <= -2.25e-267)
		tmp = Float64(x / z);
	elseif (y <= 2.75e-266)
		tmp = Float64(-x);
	elseif (y <= 4.2e-95)
		tmp = Float64(x / z);
	elseif (y <= 5.2e-78)
		tmp = Float64(-x);
	elseif (y <= 1.0)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x / z);
	tmp = 0.0;
	if (y <= -5.6e+36)
		tmp = t_0;
	elseif (y <= -7.2e-130)
		tmp = -x;
	elseif (y <= -1.45e-162)
		tmp = x / z;
	elseif (y <= -4.1e-199)
		tmp = -x;
	elseif (y <= -2.25e-267)
		tmp = x / z;
	elseif (y <= 2.75e-266)
		tmp = -x;
	elseif (y <= 4.2e-95)
		tmp = x / z;
	elseif (y <= 5.2e-78)
		tmp = -x;
	elseif (y <= 1.0)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.6e+36], t$95$0, If[LessEqual[y, -7.2e-130], (-x), If[LessEqual[y, -1.45e-162], N[(x / z), $MachinePrecision], If[LessEqual[y, -4.1e-199], (-x), If[LessEqual[y, -2.25e-267], N[(x / z), $MachinePrecision], If[LessEqual[y, 2.75e-266], (-x), If[LessEqual[y, 4.2e-95], N[(x / z), $MachinePrecision], If[LessEqual[y, 5.2e-78], (-x), If[LessEqual[y, 1.0], N[(x / z), $MachinePrecision], t$95$0]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.6000000000000001e36 or 1 < y

   1. Initial program 85.5%

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

   3. Taylor expanded in y around inf 76.6%

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

      1. associate-/l* 76.8%

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

      2. associate-/r/ 80.4%

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

   5. Simplified 80.4%

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

   if -5.6000000000000001e36 < y < -7.2000000000000003e-130 or -1.4500000000000001e-162 < y < -4.10000000000000022e-199 or -2.25e-267 < y < 2.75000000000000013e-266 or 4.2e-95 < y < 5.2000000000000002e-78

   1. Initial program 79.4%

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

   3. Taylor expanded in z around inf 68.0%

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

      1. mul-1-neg 68.0%

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

   5. Simplified 68.0%

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

   if -7.2000000000000003e-130 < y < -1.4500000000000001e-162 or -4.10000000000000022e-199 < y < -2.25e-267 or 2.75000000000000013e-266 < y < 4.2e-95 or 5.2000000000000002e-78 < y < 1

   1. Initial program 95.7%

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

   3. Taylor expanded in y around 0 95.3%

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

   4. Taylor expanded in z around 0 72.7%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+36}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-130}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-162}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-199}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-267}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-266}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-78}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{if}\;y \leq -3:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+192}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ -1.0 (/ y z)))))
   (if (<= y -3.0)
     t_0
     (if (<= y 1.0) (- (/ x z) x) (if (<= y 8.8e+192) t_0 (* y (/ x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (-1.0 + (y / z));
	double tmp;
	if (y <= -3.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = (x / z) - x;
	} else if (y <= 8.8e+192) {
		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) :: tmp
    t_0 = x * ((-1.0d0) + (y / z))
    if (y <= (-3.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = (x / z) - x
    else if (y <= 8.8d+192) 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 * (-1.0 + (y / z));
	double tmp;
	if (y <= -3.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = (x / z) - x;
	} else if (y <= 8.8e+192) {
		tmp = t_0;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (-1.0 + (y / z))
	tmp = 0
	if y <= -3.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = (x / z) - x
	elif y <= 8.8e+192:
		tmp = t_0
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-1.0 + Float64(y / z)))
	tmp = 0.0
	if (y <= -3.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(Float64(x / z) - x);
	elseif (y <= 8.8e+192)
		tmp = t_0;
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (-1.0 + (y / z));
	tmp = 0.0;
	if (y <= -3.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = (x / z) - x;
	elseif (y <= 8.8e+192)
		tmp = t_0;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.0], t$95$0, If[LessEqual[y, 1.0], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[y, 8.8e+192], t$95$0, N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3 or 1 < y < 8.8000000000000003e192

   1. Initial program 82.2%

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

   3. Taylor expanded in x around 0 82.2%

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

      1. associate--l+ 82.2%

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

      2. +-commutative 82.2%

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

      3. associate-*r/ 94.3%

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

      4. +-commutative 94.3%

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

      5. associate--l+ 94.3%

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

      6. div-sub 94.3%

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

      7. sub-neg 94.3%

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

      8. *-inverses 94.3%

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

      9. metadata-eval 94.3%

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

   5. Simplified 94.3%

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

   6. Taylor expanded in y around inf 93.7%

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

   if -3 < y < 1

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0 88.4%

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

      1. associate--l+ 88.4%

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

      2. +-commutative 88.4%

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

      3. associate-*r/ 99.7%

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

      4. +-commutative 99.7%

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

      5. associate--l+ 99.7%

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

      6. div-sub 99.7%

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

      7. sub-neg 99.7%

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

      8. *-inverses 99.7%

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

      9. metadata-eval 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 99.1%

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

      1. sub-neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. distribute-rgt-in 99.1%

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

      4. associate-*l/ 99.3%

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

      5. *-lft-identity 99.3%

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

      6. neg-mul-1 99.3%

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

      7. unsub-neg 99.3%

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

   8. Simplified 99.3%

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

   if 8.8000000000000003e192 < y

   1. Initial program 95.7%

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

   3. Taylor expanded in y around inf 94.4%

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

      1. associate-/l* 83.5%

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

      2. associate-/r/ 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+192}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-14} \lor \neg \left(z \leq 9.5 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{1 + y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5e-14) (not (<= z 9.5e-17)))
   (* x (+ -1.0 (/ (+ 1.0 y) z)))
   (/ (+ x (* x y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5e-14) || !(z <= 9.5e-17)) {
		tmp = x * (-1.0 + ((1.0 + y) / z));
	} else {
		tmp = (x + (x * y)) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-5d-14)) .or. (.not. (z <= 9.5d-17))) then
        tmp = x * ((-1.0d0) + ((1.0d0 + y) / z))
    else
        tmp = (x + (x * y)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5e-14) || !(z <= 9.5e-17)) {
		tmp = x * (-1.0 + ((1.0 + y) / z));
	} else {
		tmp = (x + (x * y)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5e-14) or not (z <= 9.5e-17):
		tmp = x * (-1.0 + ((1.0 + y) / z))
	else:
		tmp = (x + (x * y)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5e-14) || !(z <= 9.5e-17))
		tmp = Float64(x * Float64(-1.0 + Float64(Float64(1.0 + y) / z)));
	else
		tmp = Float64(Float64(x + Float64(x * y)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5e-14) || ~((z <= 9.5e-17)))
		tmp = x * (-1.0 + ((1.0 + y) / z));
	else
		tmp = (x + (x * y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5e-14], N[Not[LessEqual[z, 9.5e-17]], $MachinePrecision]], N[(x * N[(-1.0 + N[(N[(1.0 + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.0000000000000002e-14 or 9.50000000000000029e-17 < z

   1. Initial program 73.8%

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

   3. Taylor expanded in x around 0 73.8%

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

      1. associate--l+ 73.8%

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

      2. +-commutative 73.8%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--l+ 99.8%

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

      6. div-sub 99.8%

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

      7. sub-neg 99.8%

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

      8. *-inverses 99.8%

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

      9. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

   if -5.0000000000000002e-14 < z < 9.50000000000000029e-17

   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%

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

   5. Taylor expanded in z around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-14} \lor \neg \left(z \leq 9.5 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{1 + y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.0) (not (<= y 1.0))) (- (/ x (/ z y)) x) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.0) || !(y <= 1.0)) {
		tmp = (x / (z / y)) - x;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-3.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = (x / (z / y)) - x
    else
        tmp = (x / z) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.0) || !(y <= 1.0)) {
		tmp = (x / (z / y)) - x;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.0) or not (y <= 1.0):
		tmp = (x / (z / y)) - x
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.0) || !(y <= 1.0))
		tmp = Float64(Float64(x / Float64(z / y)) - x);
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.0) || ~((y <= 1.0)))
		tmp = (x / (z / y)) - x;
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3 or 1 < y

   1. Initial program 84.8%

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

   3. Taylor expanded in x around 0 84.8%

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

      1. associate--l+ 84.8%

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

      2. +-commutative 84.8%

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

      3. associate-*r/ 91.6%

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

      4. +-commutative 91.6%

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

      5. associate--l+ 91.6%

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

      6. div-sub 91.6%

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

      7. sub-neg 91.6%

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

      8. *-inverses 91.6%

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

      9. metadata-eval 91.6%

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

   5. Simplified 91.6%

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

      1. distribute-rgt-in 91.6%

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

      2. neg-mul-1 91.6%

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

      3. unsub-neg 91.6%

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

      4. clear-num 91.5%

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

      5. associate-*l/ 92.5%

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

      6. *-un-lft-identity 92.5%

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

   7. Applied egg-rr 92.5%

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

   8. Taylor expanded in y around inf 92.0%

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

   if -3 < y < 1

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0 88.4%

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

      1. associate--l+ 88.4%

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

      2. +-commutative 88.4%

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

      3. associate-*r/ 99.7%

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

      4. +-commutative 99.7%

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

      5. associate--l+ 99.7%

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

      6. div-sub 99.7%

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

      7. sub-neg 99.7%

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

      8. *-inverses 99.7%

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

      9. metadata-eval 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 99.1%

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

      1. sub-neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. distribute-rgt-in 99.1%

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

      4. associate-*l/ 99.3%

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

      5. *-lft-identity 99.3%

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

      6. neg-mul-1 99.3%

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

      7. unsub-neg 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 69.8%

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

   3. Taylor expanded in x around 0 69.8%

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

      1. associate--l+ 69.8%

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

      2. +-commutative 69.8%

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

      3. associate-*r/ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate--l+ 99.9%

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

      6. div-sub 99.9%

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

      7. sub-neg 99.9%

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

      8. *-inverses 99.9%

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

      9. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around inf 99.3%

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

   if -1 < z < 1

   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%

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

   5. Taylor expanded in z around 0 98.9%

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

   if 1 < z

   1. Initial program 73.2%

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

   3. Taylor expanded in x around 0 73.2%

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

      1. associate--l+ 73.2%

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

      2. +-commutative 73.2%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--l+ 99.8%

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

      6. div-sub 99.8%

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

      7. sub-neg 99.8%

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

      8. *-inverses 99.8%

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

      9. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. neg-mul-1 99.8%

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

      3. unsub-neg 99.8%

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

      4. clear-num 99.8%

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

      5. associate-*l/ 99.9%

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

      6. *-un-lft-identity 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in y around inf 98.3%

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

4. Final simplification 98.8%

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

Alternative 8: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+34} \lor \neg \left(y \leq 1.1 \cdot 10^{+117}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.45e+34) (not (<= y 1.1e+117))) (* y (/ x z)) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.45e+34) || !(y <= 1.1e+117)) {
		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 <= (-1.45d+34)) .or. (.not. (y <= 1.1d+117))) 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 <= -1.45e+34) || !(y <= 1.1e+117)) {
		tmp = y * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.45e+34) or not (y <= 1.1e+117):
		tmp = y * (x / z)
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.45e+34) || !(y <= 1.1e+117))
		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 <= -1.45e+34) || ~((y <= 1.1e+117)))
		tmp = y * (x / z);
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.45e+34], N[Not[LessEqual[y, 1.1e+117]], $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 < -1.4500000000000001e34 or 1.10000000000000007e117 < y

   1. Initial program 84.9%

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

   3. Taylor expanded in y around inf 77.5%

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

      1. associate-/l* 77.7%

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

      2. associate-/r/ 82.6%

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

   5. Simplified 82.6%

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

   if -1.4500000000000001e34 < y < 1.10000000000000007e117

   1. Initial program 87.9%

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

   3. Taylor expanded in x around 0 87.9%

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

      1. associate--l+ 87.9%

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

      2. +-commutative 87.9%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--l+ 99.8%

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

      6. div-sub 99.8%

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

      7. sub-neg 99.8%

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

      8. *-inverses 99.8%

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

      9. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 95.3%

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

      1. sub-neg 95.3%

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

      2. metadata-eval 95.3%

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

      3. distribute-rgt-in 95.3%

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

      4. associate-*l/ 95.6%

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

      5. *-lft-identity 95.6%

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

      6. neg-mul-1 95.6%

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

      7. unsub-neg 95.6%

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

   8. Simplified 95.6%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+34} \lor \neg \left(y \leq 1.1 \cdot 10^{+117}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+34}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.3e+34)
   (/ y (/ z x))
   (if (<= y 2.2e+116) (- (/ x z) x) (* y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.3e+34) {
		tmp = y / (z / x);
	} else if (y <= 2.2e+116) {
		tmp = (x / z) - x;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.3d+34)) then
        tmp = y / (z / x)
    else if (y <= 2.2d+116) then
        tmp = (x / z) - x
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.3e+34) {
		tmp = y / (z / x);
	} else if (y <= 2.2e+116) {
		tmp = (x / z) - x;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.3e+34:
		tmp = y / (z / x)
	elif y <= 2.2e+116:
		tmp = (x / z) - x
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.3e+34)
		tmp = Float64(y / Float64(z / x));
	elseif (y <= 2.2e+116)
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.3e+34)
		tmp = y / (z / x);
	elseif (y <= 2.2e+116)
		tmp = (x / z) - x;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.3e+34], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+116], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2999999999999998e34

   1. Initial program 81.0%

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

   3. Taylor expanded in x around 0 81.0%

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

      1. associate--l+ 81.0%

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

      2. +-commutative 81.0%

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

      3. associate-*r/ 94.1%

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

      4. +-commutative 94.1%

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

      5. associate--l+ 94.1%

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

      6. div-sub 94.1%

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

      7. sub-neg 94.1%

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

      8. *-inverses 94.1%

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

      9. metadata-eval 94.1%

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

   5. Simplified 94.1%

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

   6. Taylor expanded in y around inf 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-/l* 79.1%

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

   8. Simplified 79.1%

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

   if -2.2999999999999998e34 < y < 2.2e116

   1. Initial program 87.9%

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

   3. Taylor expanded in x around 0 87.9%

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

      1. associate--l+ 87.9%

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

      2. +-commutative 87.9%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--l+ 99.8%

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

      6. div-sub 99.8%

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

      7. sub-neg 99.8%

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

      8. *-inverses 99.8%

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

      9. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 95.3%

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

      1. sub-neg 95.3%

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

      2. metadata-eval 95.3%

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

      3. distribute-rgt-in 95.3%

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

      4. associate-*l/ 95.6%

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

      5. *-lft-identity 95.6%

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

      6. neg-mul-1 95.6%

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

      7. unsub-neg 95.6%

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

   8. Simplified 95.6%

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

   if 2.2e116 < y

   1. Initial program 90.8%

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

   3. Taylor expanded in y around inf 87.8%

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

      1. associate-/l* 79.7%

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

      2. associate-/r/ 87.9%

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

   5. Simplified 87.9%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+34}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.2e+37)
   (/ y (/ z x))
   (if (<= y 4.8e+16) (- (/ x z) x) (/ (* x y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.2e+37) {
		tmp = y / (z / x);
	} else if (y <= 4.8e+16) {
		tmp = (x / z) - x;
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.2d+37)) then
        tmp = y / (z / x)
    else if (y <= 4.8d+16) then
        tmp = (x / z) - x
    else
        tmp = (x * y) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.2e+37) {
		tmp = y / (z / x);
	} else if (y <= 4.8e+16) {
		tmp = (x / z) - x;
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.2e+37:
		tmp = y / (z / x)
	elif y <= 4.8e+16:
		tmp = (x / z) - x
	else:
		tmp = (x * y) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.2e+37)
		tmp = Float64(y / Float64(z / x));
	elseif (y <= 4.8e+16)
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = Float64(Float64(x * y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.2e+37)
		tmp = y / (z / x);
	elseif (y <= 4.8e+16)
		tmp = (x / z) - x;
	else
		tmp = (x * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.2e+37], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+16], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e37

   1. Initial program 81.0%

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

   3. Taylor expanded in x around 0 81.0%

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

      1. associate--l+ 81.0%

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

      2. +-commutative 81.0%

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

      3. associate-*r/ 94.1%

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

      4. +-commutative 94.1%

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

      5. associate--l+ 94.1%

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

      6. div-sub 94.1%

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

      7. sub-neg 94.1%

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

      8. *-inverses 94.1%

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

      9. metadata-eval 94.1%

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

   5. Simplified 94.1%

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

   6. Taylor expanded in y around inf 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-/l* 79.1%

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

   8. Simplified 79.1%

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

   if -1.2e37 < y < 4.8e16

   1. Initial program 87.3%

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

   3. Taylor expanded in x around 0 87.3%

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

      1. associate--l+ 87.3%

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

      2. +-commutative 87.3%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--l+ 99.8%

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

      6. div-sub 99.8%

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

      7. sub-neg 99.8%

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

      8. *-inverses 99.8%

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

      9. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 96.4%

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

      1. sub-neg 96.4%

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

      2. metadata-eval 96.4%

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

      3. distribute-rgt-in 96.4%

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

      4. associate-*l/ 96.6%

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

      5. *-lft-identity 96.6%

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

      6. neg-mul-1 96.6%

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

      7. unsub-neg 96.6%

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

   8. Simplified 96.6%

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

   if 4.8e16 < y

   1. Initial program 92.1%

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

   3. Taylor expanded in y around inf 87.6%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 35000000000000:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + y}} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 35000000000000.0)
   (- (/ (* x (+ 1.0 y)) z) x)
   (- (/ x (/ z (+ 1.0 y))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 35000000000000.0) {
		tmp = ((x * (1.0 + y)) / z) - x;
	} else {
		tmp = (x / (z / (1.0 + y))) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 35000000000000.0d0) then
        tmp = ((x * (1.0d0 + y)) / z) - x
    else
        tmp = (x / (z / (1.0d0 + y))) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 35000000000000.0) {
		tmp = ((x * (1.0 + y)) / z) - x;
	} else {
		tmp = (x / (z / (1.0 + y))) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 35000000000000.0:
		tmp = ((x * (1.0 + y)) / z) - x
	else:
		tmp = (x / (z / (1.0 + y))) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 35000000000000.0)
		tmp = Float64(Float64(Float64(x * Float64(1.0 + y)) / z) - x);
	else
		tmp = Float64(Float64(x / Float64(z / Float64(1.0 + y))) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 35000000000000.0)
		tmp = ((x * (1.0 + y)) / z) - x;
	else
		tmp = (x / (z / (1.0 + y))) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 35000000000000.0], N[(N[(N[(x * N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / N[(z / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5e13

   1. Initial program 92.4%

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

   3. Taylor expanded in z around 0 95.9%

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

   if 3.5e13 < x

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0 70.1%

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

      1. associate--l+ 70.1%

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

      2. +-commutative 70.1%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--l+ 99.8%

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

      6. div-sub 99.8%

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

      7. sub-neg 99.8%

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

      8. *-inverses 99.8%

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

      9. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. neg-mul-1 99.8%

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

      3. unsub-neg 99.8%

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

      4. clear-num 99.9%

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

      5. associate-*l/ 100.0%

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

      6. *-un-lft-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 35000000000000:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + y}} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-54}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + y}} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6e-54) (/ (* x (+ 1.0 (- y z))) z) (- (/ x (/ z (+ 1.0 y))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6e-54) {
		tmp = (x * (1.0 + (y - z))) / z;
	} else {
		tmp = (x / (z / (1.0 + y))) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 6d-54) then
        tmp = (x * (1.0d0 + (y - z))) / z
    else
        tmp = (x / (z / (1.0d0 + y))) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 6e-54) {
		tmp = (x * (1.0 + (y - z))) / z;
	} else {
		tmp = (x / (z / (1.0 + y))) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 6e-54:
		tmp = (x * (1.0 + (y - z))) / z
	else:
		tmp = (x / (z / (1.0 + y))) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6e-54)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(y - z))) / z);
	else
		tmp = Float64(Float64(x / Float64(z / Float64(1.0 + y))) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 6e-54)
		tmp = (x * (1.0 + (y - z))) / z;
	else
		tmp = (x / (z / (1.0 + y))) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6e-54], N[(N[(x * N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / N[(z / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.00000000000000018e-54

   1. Initial program 92.0%

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

   if 6.00000000000000018e-54 < x

   1. Initial program 74.0%

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

   3. Taylor expanded in x around 0 74.0%

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

      1. associate--l+ 74.0%

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

      2. +-commutative 74.0%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--l+ 99.8%

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

      6. div-sub 99.8%

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

      7. sub-neg 99.8%

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

      8. *-inverses 99.8%

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

      9. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. neg-mul-1 99.8%

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

      3. unsub-neg 99.8%

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

      4. clear-num 99.8%

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

      5. associate-*l/ 100.0%

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

      6. *-un-lft-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-54}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + y}} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.014 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.014) (not (<= z 1.0))) (- x) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.014) || !(z <= 1.0)) {
		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 <= (-0.014d0)) .or. (.not. (z <= 1.0d0))) 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 <= -0.014) || !(z <= 1.0)) {
		tmp = -x;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -0.014) or not (z <= 1.0):
		tmp = -x
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.014) || !(z <= 1.0))
		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 <= -0.014) || ~((z <= 1.0)))
		tmp = -x;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.014], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], (-x), N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0140000000000000003 or 1 < z

   1. Initial program 72.1%

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

   3. Taylor expanded in z around inf 65.6%

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

      1. mul-1-neg 65.6%

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

   5. Simplified 65.6%

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

   if -0.0140000000000000003 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 57.8%

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

   4. Taylor expanded in z around 0 56.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.014 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.2% 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 86.6%

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

3. Taylor expanded in z around inf 32.8%

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

   1. mul-1-neg 32.8%

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

5. Simplified 32.8%

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

6. Final simplification 32.8%

   \[\leadsto -x \]
7. Add Preprocessing

Developer target: 99.5% accurate, 0.4× 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 2024041 
(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))