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

Percentage Accurate: 87.7% → 99.8%

Time: 8.7s

Alternatives: 14

Speedup: 1.0×

• 20.7% 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: 87.7% 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, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 7.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{x\_m \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{y + 1}} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 7.5e-54)
    (/ (* x_m (+ (- y z) 1.0)) z)
    (- (/ x_m (/ z (+ y 1.0))) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 7.5e-54) {
		tmp = (x_m * ((y - z) + 1.0)) / z;
	} else {
		tmp = (x_m / (z / (y + 1.0))) - x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 7.5d-54) then
        tmp = (x_m * ((y - z) + 1.0d0)) / z
    else
        tmp = (x_m / (z / (y + 1.0d0))) - x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 7.5e-54) {
		tmp = (x_m * ((y - z) + 1.0)) / z;
	} else {
		tmp = (x_m / (z / (y + 1.0))) - x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 7.5e-54:
		tmp = (x_m * ((y - z) + 1.0)) / z
	else:
		tmp = (x_m / (z / (y + 1.0))) - x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 7.5e-54)
		tmp = Float64(Float64(x_m * Float64(Float64(y - z) + 1.0)) / z);
	else
		tmp = Float64(Float64(x_m / Float64(z / Float64(y + 1.0))) - x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 7.5e-54)
		tmp = (x_m * ((y - z) + 1.0)) / z;
	else
		tmp = (x_m / (z / (y + 1.0))) - x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 7.5e-54], N[(N[(x$95$m * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / N[(z / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.5000000000000005e-54

   1. Initial program 90.0%

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

   if 7.5000000000000005e-54 < x

   1. Initial program 76.4%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 99.9%

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

      5. *-inverses 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. distribute-lft-in 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

      4. *-commutative 99.9%

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

      5. mul-1-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + 1}} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+14}:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-83}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -1.45e+14)
    (- x_m)
    (if (<= z 4.8e-178)
      (* y (/ x_m z))
      (if (<= z 6.6e-83)
        (/ x_m z)
        (if (<= z 1.4e+77) (* x_m (/ y z)) (- x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1.45e+14) {
		tmp = -x_m;
	} else if (z <= 4.8e-178) {
		tmp = y * (x_m / z);
	} else if (z <= 6.6e-83) {
		tmp = x_m / z;
	} else if (z <= 1.4e+77) {
		tmp = x_m * (y / z);
	} else {
		tmp = -x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.45d+14)) then
        tmp = -x_m
    else if (z <= 4.8d-178) then
        tmp = y * (x_m / z)
    else if (z <= 6.6d-83) then
        tmp = x_m / z
    else if (z <= 1.4d+77) then
        tmp = x_m * (y / z)
    else
        tmp = -x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1.45e+14) {
		tmp = -x_m;
	} else if (z <= 4.8e-178) {
		tmp = y * (x_m / z);
	} else if (z <= 6.6e-83) {
		tmp = x_m / z;
	} else if (z <= 1.4e+77) {
		tmp = x_m * (y / z);
	} else {
		tmp = -x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= -1.45e+14:
		tmp = -x_m
	elif z <= 4.8e-178:
		tmp = y * (x_m / z)
	elif z <= 6.6e-83:
		tmp = x_m / z
	elif z <= 1.4e+77:
		tmp = x_m * (y / z)
	else:
		tmp = -x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -1.45e+14)
		tmp = Float64(-x_m);
	elseif (z <= 4.8e-178)
		tmp = Float64(y * Float64(x_m / z));
	elseif (z <= 6.6e-83)
		tmp = Float64(x_m / z);
	elseif (z <= 1.4e+77)
		tmp = Float64(x_m * Float64(y / z));
	else
		tmp = Float64(-x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -1.45e+14)
		tmp = -x_m;
	elseif (z <= 4.8e-178)
		tmp = y * (x_m / z);
	elseif (z <= 6.6e-83)
		tmp = x_m / z;
	elseif (z <= 1.4e+77)
		tmp = x_m * (y / z);
	else
		tmp = -x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -1.45e+14], (-x$95$m), If[LessEqual[z, 4.8e-178], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.6e-83], N[(x$95$m / z), $MachinePrecision], If[LessEqual[z, 1.4e+77], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], (-x$95$m)]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -1.45e14 or 1.4e77 < z

   1. Initial program 66.9%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 100.0%

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

      5. *-inverses 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 82.0%

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

      1. neg-mul-1 82.0%

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

   7. Simplified 82.0%

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

   if -1.45e14 < z < 4.8000000000000001e-178

   1. Initial program 99.9%

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

      1. associate-/l* 93.1%

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

      2. +-commutative 93.1%

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

      3. associate-+r- 93.1%

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

      4. div-sub 93.1%

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

      5. *-inverses 93.1%

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

      6. sub-neg 93.1%

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

      7. metadata-eval 93.1%

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

      8. +-commutative 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in y around inf 61.9%

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

      1. associate-*l/ 64.1%

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

   7. Applied egg-rr 64.1%

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

   if 4.8000000000000001e-178 < z < 6.5999999999999999e-83

   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-define 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%

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

   6. Taylor expanded in y around 0 77.9%

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

   if 6.5999999999999999e-83 < z < 1.4e77

   1. Initial program 99.7%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+r- 99.5%

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

      4. div-sub 99.5%

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

      5. *-inverses 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

      8. +-commutative 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 63.1%

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

      1. associate-/l* 62.9%

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

   7. Simplified 62.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+14}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-84}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+79}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -1.0)
    (- x_m)
    (if (<= z 3.8e-84)
      (/ x_m z)
      (if (<= z 1.35e+79) (* x_m (/ y z)) (- x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = -x_m;
	} else if (z <= 3.8e-84) {
		tmp = x_m / z;
	} else if (z <= 1.35e+79) {
		tmp = x_m * (y / z);
	} else {
		tmp = -x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = -x_m
    else if (z <= 3.8d-84) then
        tmp = x_m / z
    else if (z <= 1.35d+79) then
        tmp = x_m * (y / z)
    else
        tmp = -x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = -x_m;
	} else if (z <= 3.8e-84) {
		tmp = x_m / z;
	} else if (z <= 1.35e+79) {
		tmp = x_m * (y / z);
	} else {
		tmp = -x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = -x_m
	elif z <= 3.8e-84:
		tmp = x_m / z
	elif z <= 1.35e+79:
		tmp = x_m * (y / z)
	else:
		tmp = -x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(-x_m);
	elseif (z <= 3.8e-84)
		tmp = Float64(x_m / z);
	elseif (z <= 1.35e+79)
		tmp = Float64(x_m * Float64(y / z));
	else
		tmp = Float64(-x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = -x_m;
	elseif (z <= 3.8e-84)
		tmp = x_m / z;
	elseif (z <= 1.35e+79)
		tmp = x_m * (y / z);
	else
		tmp = -x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -1.0], (-x$95$m), If[LessEqual[z, 3.8e-84], N[(x$95$m / z), $MachinePrecision], If[LessEqual[z, 1.35e+79], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], (-x$95$m)]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 1.35e79 < z

   1. Initial program 68.3%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 99.9%

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

      5. *-inverses 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 79.6%

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

      1. neg-mul-1 79.6%

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

   7. Simplified 79.6%

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

   if -1 < z < 3.79999999999999986e-84

   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-define 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 99.3%

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

   6. Taylor expanded in y around 0 62.6%

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

   if 3.79999999999999986e-84 < z < 1.35e79

   1. Initial program 99.7%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+r- 99.5%

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

      4. div-sub 99.5%

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

      5. *-inverses 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

      8. +-commutative 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 63.1%

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

      1. associate-/l* 62.9%

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

   7. Simplified 62.9%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+79}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{+26} \lor \neg \left(y \leq 3.2 \cdot 10^{-14}\right):\\ \;\;\;\;x\_m \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -1.52e+26) (not (<= y 3.2e-14)))
    (* x_m (+ -1.0 (/ y z)))
    (- (/ x_m z) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -1.52e+26) || !(y <= 3.2e-14)) {
		tmp = x_m * (-1.0 + (y / z));
	} else {
		tmp = (x_m / z) - x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.52d+26)) .or. (.not. (y <= 3.2d-14))) then
        tmp = x_m * ((-1.0d0) + (y / z))
    else
        tmp = (x_m / z) - x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -1.52e+26) || !(y <= 3.2e-14)) {
		tmp = x_m * (-1.0 + (y / z));
	} else {
		tmp = (x_m / z) - x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (y <= -1.52e+26) or not (y <= 3.2e-14):
		tmp = x_m * (-1.0 + (y / z))
	else:
		tmp = (x_m / z) - x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((y <= -1.52e+26) || !(y <= 3.2e-14))
		tmp = Float64(x_m * Float64(-1.0 + Float64(y / z)));
	else
		tmp = Float64(Float64(x_m / z) - x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((y <= -1.52e+26) || ~((y <= 3.2e-14)))
		tmp = x_m * (-1.0 + (y / z));
	else
		tmp = (x_m / z) - x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[y, -1.52e+26], N[Not[LessEqual[y, 3.2e-14]], $MachinePrecision]], N[(x$95$m * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.52e26 or 3.2000000000000002e-14 < y

   1. Initial program 83.7%

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

      1. associate-/l* 95.6%

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

      2. +-commutative 95.6%

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

      3. associate-+r- 95.6%

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

      4. div-sub 95.6%

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

      5. *-inverses 95.6%

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

      6. sub-neg 95.6%

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

      7. metadata-eval 95.6%

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

      8. +-commutative 95.6%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in y around inf 95.6%

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

   if -1.52e26 < y < 3.2000000000000002e-14

   1. Initial program 87.6%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 99.8%

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

      5. *-inverses 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. associate-*l/ 99.9%

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

      5. *-lft-identity 99.9%

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

      6. neg-mul-1 99.9%

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

      7. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{+26} \lor \neg \left(y \leq 3.2 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{+26} \lor \neg \left(y \leq 3.2 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{x\_m}{\frac{z}{y}} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -1.52e+26) (not (<= y 3.2e-14)))
    (- (/ x_m (/ z y)) x_m)
    (- (/ x_m z) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -1.52e+26) || !(y <= 3.2e-14)) {
		tmp = (x_m / (z / y)) - x_m;
	} else {
		tmp = (x_m / z) - x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.52d+26)) .or. (.not. (y <= 3.2d-14))) then
        tmp = (x_m / (z / y)) - x_m
    else
        tmp = (x_m / z) - x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -1.52e+26) || !(y <= 3.2e-14)) {
		tmp = (x_m / (z / y)) - x_m;
	} else {
		tmp = (x_m / z) - x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (y <= -1.52e+26) or not (y <= 3.2e-14):
		tmp = (x_m / (z / y)) - x_m
	else:
		tmp = (x_m / z) - x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((y <= -1.52e+26) || !(y <= 3.2e-14))
		tmp = Float64(Float64(x_m / Float64(z / y)) - x_m);
	else
		tmp = Float64(Float64(x_m / z) - x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((y <= -1.52e+26) || ~((y <= 3.2e-14)))
		tmp = (x_m / (z / y)) - x_m;
	else
		tmp = (x_m / z) - x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[y, -1.52e+26], N[Not[LessEqual[y, 3.2e-14]], $MachinePrecision]], N[(N[(x$95$m / N[(z / y), $MachinePrecision]), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.52e26 or 3.2000000000000002e-14 < y

   1. Initial program 83.7%

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

      1. associate-/l* 95.6%

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

      2. +-commutative 95.6%

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

      3. associate-+r- 95.6%

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

      4. div-sub 95.6%

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

      5. *-inverses 95.6%

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

      6. sub-neg 95.6%

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

      7. metadata-eval 95.6%

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

      8. +-commutative 95.6%

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

   3. Simplified 95.6%

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

      1. distribute-lft-in 95.6%

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

      2. clear-num 95.6%

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

      3. un-div-inv 96.5%

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

      4. *-commutative 96.5%

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

      5. mul-1-neg 96.5%

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

   6. Applied egg-rr 96.5%

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

   7. Taylor expanded in y around inf 96.5%

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

   if -1.52e26 < y < 3.2000000000000002e-14

   1. Initial program 87.6%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 99.8%

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

      5. *-inverses 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. associate-*l/ 99.9%

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

      5. *-lft-identity 99.9%

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

      6. neg-mul-1 99.9%

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

      7. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{+26} \lor \neg \left(y \leq 3.2 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{+26}:\\ \;\;\;\;x\_m \cdot \frac{y}{z} - x\_m\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(-1 + \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -1.52e+26)
    (- (* x_m (/ y z)) x_m)
    (if (<= y 3.2e-14) (- (/ x_m z) x_m) (* x_m (+ -1.0 (/ y z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -1.52e+26) {
		tmp = (x_m * (y / z)) - x_m;
	} else if (y <= 3.2e-14) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = x_m * (-1.0 + (y / z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.52d+26)) then
        tmp = (x_m * (y / z)) - x_m
    else if (y <= 3.2d-14) then
        tmp = (x_m / z) - x_m
    else
        tmp = x_m * ((-1.0d0) + (y / z))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -1.52e+26) {
		tmp = (x_m * (y / z)) - x_m;
	} else if (y <= 3.2e-14) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = x_m * (-1.0 + (y / z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -1.52e+26:
		tmp = (x_m * (y / z)) - x_m
	elif y <= 3.2e-14:
		tmp = (x_m / z) - x_m
	else:
		tmp = x_m * (-1.0 + (y / z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -1.52e+26)
		tmp = Float64(Float64(x_m * Float64(y / z)) - x_m);
	elseif (y <= 3.2e-14)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(x_m * Float64(-1.0 + Float64(y / z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -1.52e+26)
		tmp = (x_m * (y / z)) - x_m;
	elseif (y <= 3.2e-14)
		tmp = (x_m / z) - x_m;
	else
		tmp = x_m * (-1.0 + (y / z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -1.52e+26], N[(N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision] - x$95$m), $MachinePrecision], If[LessEqual[y, 3.2e-14], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(x$95$m * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.52e26

   1. Initial program 83.6%

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

      1. associate-/l* 92.3%

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

      2. +-commutative 92.3%

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

      3. associate-+r- 92.3%

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

      4. div-sub 92.3%

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

      5. *-inverses 92.3%

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

      6. sub-neg 92.3%

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

      7. metadata-eval 92.3%

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

      8. +-commutative 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in y around inf 92.3%

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

      1. distribute-rgt-in 92.4%

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

      2. neg-mul-1 92.4%

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

      3. unsub-neg 92.4%

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

      4. *-commutative 92.4%

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

   7. Applied egg-rr 92.4%

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

   if -1.52e26 < y < 3.2000000000000002e-14

   1. Initial program 87.6%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 99.8%

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

      5. *-inverses 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. associate-*l/ 99.9%

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

      5. *-lft-identity 99.9%

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

      6. neg-mul-1 99.9%

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

      7. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   if 3.2000000000000002e-14 < y

   1. Initial program 83.8%

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

      1. associate-/l* 98.5%

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

      2. +-commutative 98.5%

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

      3. associate-+r- 98.5%

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

      4. div-sub 98.5%

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

      5. *-inverses 98.5%

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

      6. sub-neg 98.5%

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

      7. metadata-eval 98.5%

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

      8. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in y around inf 98.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{+26}:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.1:\\ \;\;\;\;x\_m \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x\_m + x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y}{z} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -1.1)
    (* x_m (+ -1.0 (/ y z)))
    (if (<= z 1.0) (/ (+ x_m (* x_m y)) z) (- (* x_m (/ y z)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1.1) {
		tmp = x_m * (-1.0 + (y / z));
	} else if (z <= 1.0) {
		tmp = (x_m + (x_m * y)) / z;
	} else {
		tmp = (x_m * (y / z)) - x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.1d0)) then
        tmp = x_m * ((-1.0d0) + (y / z))
    else if (z <= 1.0d0) then
        tmp = (x_m + (x_m * y)) / z
    else
        tmp = (x_m * (y / z)) - x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1.1) {
		tmp = x_m * (-1.0 + (y / z));
	} else if (z <= 1.0) {
		tmp = (x_m + (x_m * y)) / z;
	} else {
		tmp = (x_m * (y / z)) - x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= -1.1:
		tmp = x_m * (-1.0 + (y / z))
	elif z <= 1.0:
		tmp = (x_m + (x_m * y)) / z
	else:
		tmp = (x_m * (y / z)) - x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -1.1)
		tmp = Float64(x_m * Float64(-1.0 + Float64(y / z)));
	elseif (z <= 1.0)
		tmp = Float64(Float64(x_m + Float64(x_m * y)) / z);
	else
		tmp = Float64(Float64(x_m * Float64(y / z)) - x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -1.1)
		tmp = x_m * (-1.0 + (y / z));
	elseif (z <= 1.0)
		tmp = (x_m + (x_m * y)) / z;
	else
		tmp = (x_m * (y / z)) - x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -1.1], N[(x$95$m * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(x$95$m + N[(x$95$m * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision] - x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.1000000000000001

   1. Initial program 70.1%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 100.0%

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

      5. *-inverses 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 97.1%

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

   if -1.1000000000000001 < 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-define 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 97.6%

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

   if 1 < z

   1. Initial program 73.2%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 99.9%

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

      5. *-inverses 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 99.0%

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

      1. distribute-rgt-in 99.0%

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

      2. neg-mul-1 99.0%

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

      3. unsub-neg 99.0%

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

      4. *-commutative 99.0%

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

   7. Applied egg-rr 99.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.96:\\ \;\;\;\;x\_m \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y}{z} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -0.96)
    (* x_m (+ -1.0 (/ y z)))
    (if (<= z 1.0) (* (/ x_m z) (+ y 1.0)) (- (* x_m (/ y z)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -0.96) {
		tmp = x_m * (-1.0 + (y / z));
	} else if (z <= 1.0) {
		tmp = (x_m / z) * (y + 1.0);
	} else {
		tmp = (x_m * (y / z)) - x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-0.96d0)) then
        tmp = x_m * ((-1.0d0) + (y / z))
    else if (z <= 1.0d0) then
        tmp = (x_m / z) * (y + 1.0d0)
    else
        tmp = (x_m * (y / z)) - x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -0.96) {
		tmp = x_m * (-1.0 + (y / z));
	} else if (z <= 1.0) {
		tmp = (x_m / z) * (y + 1.0);
	} else {
		tmp = (x_m * (y / z)) - x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= -0.96:
		tmp = x_m * (-1.0 + (y / z))
	elif z <= 1.0:
		tmp = (x_m / z) * (y + 1.0)
	else:
		tmp = (x_m * (y / z)) - x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -0.96)
		tmp = Float64(x_m * Float64(-1.0 + Float64(y / z)));
	elseif (z <= 1.0)
		tmp = Float64(Float64(x_m / z) * Float64(y + 1.0));
	else
		tmp = Float64(Float64(x_m * Float64(y / z)) - x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -0.96)
		tmp = x_m * (-1.0 + (y / z));
	elseif (z <= 1.0)
		tmp = (x_m / z) * (y + 1.0);
	else
		tmp = (x_m * (y / z)) - x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -0.96], N[(x$95$m * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(x$95$m / z), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision] - x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -0.95999999999999996

   1. Initial program 70.1%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 100.0%

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

      5. *-inverses 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 97.1%

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

   if -0.95999999999999996 < 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-define 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 97.6%

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

   6. Taylor expanded in x around -inf 97.6%

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

      1. mul-1-neg 97.6%

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

      2. sub-neg 97.6%

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

      3. neg-mul-1 97.6%

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

      4. metadata-eval 97.6%

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

      5. +-commutative 97.6%

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

      6. associate-*l/ 97.6%

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

      7. distribute-rgt-neg-in 97.6%

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

      8. unsub-neg 97.6%

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

   8. Simplified 97.6%

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

   if 1 < z

   1. Initial program 73.2%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 99.9%

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

      5. *-inverses 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 99.0%

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

      1. distribute-rgt-in 99.0%

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

      2. neg-mul-1 99.0%

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

      3. unsub-neg 99.0%

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

      4. *-commutative 99.0%

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

   7. Applied egg-rr 99.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.96:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+26} \lor \neg \left(y \leq 1.95 \cdot 10^{+74}\right):\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -6e+26) (not (<= y 1.95e+74)))
    (* y (/ x_m z))
    (- (/ x_m z) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -6e+26) || !(y <= 1.95e+74)) {
		tmp = y * (x_m / z);
	} else {
		tmp = (x_m / z) - x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-6d+26)) .or. (.not. (y <= 1.95d+74))) then
        tmp = y * (x_m / z)
    else
        tmp = (x_m / z) - x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -6e+26) || !(y <= 1.95e+74)) {
		tmp = y * (x_m / z);
	} else {
		tmp = (x_m / z) - x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (y <= -6e+26) or not (y <= 1.95e+74):
		tmp = y * (x_m / z)
	else:
		tmp = (x_m / z) - x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((y <= -6e+26) || !(y <= 1.95e+74))
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m / z) - x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((y <= -6e+26) || ~((y <= 1.95e+74)))
		tmp = y * (x_m / z);
	else
		tmp = (x_m / z) - x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[y, -6e+26], N[Not[LessEqual[y, 1.95e+74]], $MachinePrecision]], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -5.99999999999999994e26 or 1.95000000000000004e74 < y

   1. Initial program 85.5%

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

      1. associate-/l* 94.7%

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

      2. +-commutative 94.7%

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

      3. associate-+r- 94.7%

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

      4. div-sub 94.7%

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

      5. *-inverses 94.7%

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

      6. sub-neg 94.7%

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

      7. metadata-eval 94.7%

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

      8. +-commutative 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in y around inf 75.2%

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

      1. associate-*l/ 77.3%

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

   7. Applied egg-rr 77.3%

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

   if -5.99999999999999994e26 < y < 1.95000000000000004e74

   1. Initial program 85.6%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 99.8%

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

      5. *-inverses 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 96.6%

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

      1. sub-neg 96.6%

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

      2. metadata-eval 96.6%

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

      3. distribute-rgt-in 96.6%

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

      4. associate-*l/ 96.7%

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

      5. *-lft-identity 96.7%

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

      6. neg-mul-1 96.7%

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

      7. unsub-neg 96.7%

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

   7. Simplified 96.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+26} \lor \neg \left(y \leq 1.95 \cdot 10^{+74}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+77}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -2.8e+26)
    (/ (* x_m y) z)
    (if (<= y 1.25e+77) (- (/ x_m z) x_m) (* y (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -2.8e+26) {
		tmp = (x_m * y) / z;
	} else if (y <= 1.25e+77) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = y * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.8d+26)) then
        tmp = (x_m * y) / z
    else if (y <= 1.25d+77) then
        tmp = (x_m / z) - x_m
    else
        tmp = y * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -2.8e+26) {
		tmp = (x_m * y) / z;
	} else if (y <= 1.25e+77) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = y * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -2.8e+26:
		tmp = (x_m * y) / z
	elif y <= 1.25e+77:
		tmp = (x_m / z) - x_m
	else:
		tmp = y * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -2.8e+26)
		tmp = Float64(Float64(x_m * y) / z);
	elseif (y <= 1.25e+77)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(y * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -2.8e+26)
		tmp = (x_m * y) / z;
	elseif (y <= 1.25e+77)
		tmp = (x_m / z) - x_m;
	else
		tmp = y * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -2.8e+26], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 1.25e+77], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -2.8e26

   1. Initial program 83.6%

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

      1. associate-/l* 92.3%

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

      2. +-commutative 92.3%

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

      3. associate-+r- 92.3%

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

      4. div-sub 92.3%

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

      5. *-inverses 92.3%

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

      6. sub-neg 92.3%

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

      7. metadata-eval 92.3%

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

      8. +-commutative 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in y around inf 76.1%

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

   if -2.8e26 < y < 1.25000000000000001e77

   1. Initial program 85.6%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 99.8%

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

      5. *-inverses 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 96.6%

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

      1. sub-neg 96.6%

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

      2. metadata-eval 96.6%

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

      3. distribute-rgt-in 96.6%

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

      4. associate-*l/ 96.7%

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

      5. *-lft-identity 96.7%

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

      6. neg-mul-1 96.7%

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

      7. unsub-neg 96.7%

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

   7. Simplified 96.7%

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

   if 1.25000000000000001e77 < y

   1. Initial program 88.1%

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

      1. associate-/l* 97.9%

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

      2. +-commutative 97.9%

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

      3. associate-+r- 97.9%

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

      4. div-sub 97.9%

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

      5. *-inverses 97.9%

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

      6. sub-neg 97.9%

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

      7. metadata-eval 97.9%

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

      8. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in y around inf 74.1%

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

      1. associate-*l/ 79.0%

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

   7. Applied egg-rr 79.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 92000000000:\\ \;\;\;\;\frac{x\_m \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y + 1}{z} + -1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 92000000000.0)
    (/ (* x_m (+ (- y z) 1.0)) z)
    (* x_m (+ (/ (+ y 1.0) z) -1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 92000000000.0) {
		tmp = (x_m * ((y - z) + 1.0)) / z;
	} else {
		tmp = x_m * (((y + 1.0) / z) + -1.0);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 92000000000.0d0) then
        tmp = (x_m * ((y - z) + 1.0d0)) / z
    else
        tmp = x_m * (((y + 1.0d0) / z) + (-1.0d0))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 92000000000.0) {
		tmp = (x_m * ((y - z) + 1.0)) / z;
	} else {
		tmp = x_m * (((y + 1.0) / z) + -1.0);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 92000000000.0:
		tmp = (x_m * ((y - z) + 1.0)) / z
	else:
		tmp = x_m * (((y + 1.0) / z) + -1.0)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 92000000000.0)
		tmp = Float64(Float64(x_m * Float64(Float64(y - z) + 1.0)) / z);
	else
		tmp = Float64(x_m * Float64(Float64(Float64(y + 1.0) / z) + -1.0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 92000000000.0)
		tmp = (x_m * ((y - z) + 1.0)) / z;
	else
		tmp = x_m * (((y + 1.0) / z) + -1.0);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 92000000000.0], N[(N[(x$95$m * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * N[(N[(N[(y + 1.0), $MachinePrecision] / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.2e10

   1. Initial program 90.5%

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

   if 9.2e10 < x

   1. Initial program 73.6%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 99.9%

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

      5. *-inverses 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 92000000000:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y + 1}{z} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;-x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (or (<= z -1.0) (not (<= z 1.0))) (- x_m) (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = -x_m;
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = -x_m
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = -x_m;
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = -x_m
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(-x_m);
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = -x_m;
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], (-x$95$m), N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 71.7%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 99.9%

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

      5. *-inverses 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 75.5%

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

      1. neg-mul-1 75.5%

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

   7. Simplified 75.5%

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

   if -1 < z < 1

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. 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-define 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 97.6%

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

   6. Taylor expanded in y around 0 55.9%

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

4. Final simplification 65.9%

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

Alternative 13: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot \left(\frac{y + 1}{z} + -1\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (* x_m (+ (/ (+ y 1.0) z) -1.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m * (((y + 1.0) / z) + -1.0));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (x_m * (((y + 1.0d0) / z) + (-1.0d0)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m * (((y + 1.0) / z) + -1.0));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (x_m * (((y + 1.0) / z) + -1.0))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(x_m * Float64(Float64(Float64(y + 1.0) / z) + -1.0)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (x_m * (((y + 1.0) / z) + -1.0));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(x$95$m * N[(N[(N[(y + 1.0), $MachinePrecision] / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.6%

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

   1. associate-/l* 97.6%

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

   2. +-commutative 97.6%

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

   3. associate-+r- 97.6%

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

   4. div-sub 97.6%

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

   5. *-inverses 97.6%

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

   6. sub-neg 97.6%

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

   7. metadata-eval 97.6%

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

   8. +-commutative 97.6%

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

3. Simplified 97.6%

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

5. Final simplification 97.6%

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

Alternative 14: 38.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(-x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (- x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * -x_m;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * -x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * -x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * -x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(-x_m))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * -x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * (-x$95$m)), $MachinePrecision]

Derivation

1. Initial program 85.6%

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

   1. associate-/l* 97.6%

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

   2. +-commutative 97.6%

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

   3. associate-+r- 97.6%

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

   4. div-sub 97.6%

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

   5. *-inverses 97.6%

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

   6. sub-neg 97.6%

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

   7. metadata-eval 97.6%

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

   8. +-commutative 97.6%

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

3. Simplified 97.6%

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

5. Taylor expanded in z around inf 40.1%

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

   1. neg-mul-1 40.1%

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

7. Simplified 40.1%

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

8. Final simplification 40.1%

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

Developer target: 99.3% 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 2024047 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (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))