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

Percentage Accurate: 88.3% → 99.8%

Time: 8.5s

Alternatives: 14

Speedup: 0.6×

• 21.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.3% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\frac{x\_m \cdot \left(1 + y\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{1 + y}} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2e-48)
    (- (/ (* x_m (+ 1.0 y)) z) x_m)
    (- (/ x_m (/ z (+ 1.0 y))) 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 <= 2e-48) {
		tmp = ((x_m * (1.0 + y)) / z) - x_m;
	} else {
		tmp = (x_m / (z / (1.0 + y))) - 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 <= 2d-48) then
        tmp = ((x_m * (1.0d0 + y)) / z) - x_m
    else
        tmp = (x_m / (z / (1.0d0 + y))) - 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 <= 2e-48) {
		tmp = ((x_m * (1.0 + y)) / z) - x_m;
	} else {
		tmp = (x_m / (z / (1.0 + y))) - 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 <= 2e-48:
		tmp = ((x_m * (1.0 + y)) / z) - x_m
	else:
		tmp = (x_m / (z / (1.0 + y))) - 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 <= 2e-48)
		tmp = Float64(Float64(Float64(x_m * Float64(1.0 + y)) / z) - x_m);
	else
		tmp = Float64(Float64(x_m / Float64(z / Float64(1.0 + y))) - 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 <= 2e-48)
		tmp = ((x_m * (1.0 + y)) / z) - x_m;
	else
		tmp = (x_m / (z / (1.0 + y))) - 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, 2e-48], N[(N[(N[(x$95$m * N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(x$95$m / N[(z / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e-48

   1. Initial program 93.1%

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

      1. associate-/l* 92.4%

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

      2. +-commutative 92.4%

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

      3. associate-+r- 92.4%

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

      4. div-sub 92.4%

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

      5. *-inverses 92.4%

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

      6. sub-neg 92.4%

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

      7. +-commutative 92.4%

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

      8. metadata-eval 92.4%

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

   3. Simplified 92.4%

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

      1. distribute-lft-in 92.4%

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

      2. clear-num 92.4%

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

      3. un-div-inv 93.8%

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

      4. *-commutative 93.8%

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

      5. mul-1-neg 93.8%

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

   6. Applied egg-rr 93.8%

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

   7. Taylor expanded in z around inf 97.5%

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

   if 1.9999999999999999e-48 < x

   1. Initial program 80.8%

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

      1. associate-/l* 99.9%

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

      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. +-commutative 99.9%

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

      8. metadata-eval 99.9%

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

   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 100.0%

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

      4. *-commutative 100.0%

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

      5. mul-1-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + y}} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+45}:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-14}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-125}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-239}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -1.75e+45)
    (- x_m)
    (if (<= z -8e-14)
      (* x_m (/ y z))
      (if (<= z -2.6e-125)
        (/ x_m z)
        (if (<= z 1.4e-239)
          (* y (/ x_m z))
          (if (<= z 1.0) (/ 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 (z <= -1.75e+45) {
		tmp = -x_m;
	} else if (z <= -8e-14) {
		tmp = x_m * (y / z);
	} else if (z <= -2.6e-125) {
		tmp = x_m / z;
	} else if (z <= 1.4e-239) {
		tmp = y * (x_m / z);
	} else if (z <= 1.0) {
		tmp = x_m / 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 <= (-1.75d+45)) then
        tmp = -x_m
    else if (z <= (-8d-14)) then
        tmp = x_m * (y / z)
    else if (z <= (-2.6d-125)) then
        tmp = x_m / z
    else if (z <= 1.4d-239) then
        tmp = y * (x_m / z)
    else if (z <= 1.0d0) then
        tmp = x_m / 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 <= -1.75e+45) {
		tmp = -x_m;
	} else if (z <= -8e-14) {
		tmp = x_m * (y / z);
	} else if (z <= -2.6e-125) {
		tmp = x_m / z;
	} else if (z <= 1.4e-239) {
		tmp = y * (x_m / z);
	} else if (z <= 1.0) {
		tmp = x_m / 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 <= -1.75e+45:
		tmp = -x_m
	elif z <= -8e-14:
		tmp = x_m * (y / z)
	elif z <= -2.6e-125:
		tmp = x_m / z
	elif z <= 1.4e-239:
		tmp = y * (x_m / z)
	elif z <= 1.0:
		tmp = x_m / 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 <= -1.75e+45)
		tmp = Float64(-x_m);
	elseif (z <= -8e-14)
		tmp = Float64(x_m * Float64(y / z));
	elseif (z <= -2.6e-125)
		tmp = Float64(x_m / z);
	elseif (z <= 1.4e-239)
		tmp = Float64(y * Float64(x_m / z));
	elseif (z <= 1.0)
		tmp = Float64(x_m / 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 <= -1.75e+45)
		tmp = -x_m;
	elseif (z <= -8e-14)
		tmp = x_m * (y / z);
	elseif (z <= -2.6e-125)
		tmp = x_m / z;
	elseif (z <= 1.4e-239)
		tmp = y * (x_m / z);
	elseif (z <= 1.0)
		tmp = x_m / 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, -1.75e+45], (-x$95$m), If[LessEqual[z, -8e-14], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2.6e-125], N[(x$95$m / z), $MachinePrecision], If[LessEqual[z, 1.4e-239], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(x$95$m / z), $MachinePrecision], (-x$95$m)]]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -1.75000000000000011e45 or 1 < z

   1. Initial program 77.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. +-commutative 99.9%

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

      8. metadata-eval 99.9%

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

   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 77.0%

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

      1. neg-mul-1 77.0%

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

   7. Simplified 77.0%

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

   if -1.75000000000000011e45 < z < -7.99999999999999999e-14

   1. Initial program 99.8%

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

      1. associate-/l* 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r- 99.4%

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

      4. div-sub 99.4%

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

      5. *-inverses 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 87.9%

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

      1. associate-/l* 87.5%

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

   7. Simplified 87.5%

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

   if -7.99999999999999999e-14 < z < -2.60000000000000006e-125 or 1.40000000000000006e-239 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 65.8%

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

      1. *-commutative 65.8%

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

      2. sub-neg 65.8%

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

      3. metadata-eval 65.8%

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

      4. distribute-neg-in 65.8%

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

      5. +-commutative 65.8%

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

      6. associate-/l* 65.8%

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

      7. +-commutative 65.8%

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

      8. distribute-neg-in 65.8%

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

      9. metadata-eval 65.8%

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

      10. sub-neg 65.8%

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

   5. Simplified 65.8%

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

   6. Taylor expanded in z around 0 64.8%

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

   if -2.60000000000000006e-125 < z < 1.40000000000000006e-239

   1. Initial program 100.0%

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

      1. associate-/l* 83.5%

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

      2. +-commutative 83.5%

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

      3. associate-+r- 83.5%

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

      4. div-sub 83.5%

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

      5. *-inverses 83.5%

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

      6. sub-neg 83.5%

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

      7. +-commutative 83.5%

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

      8. metadata-eval 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in y around inf 74.0%

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

      1. *-commutative 74.0%

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

      2. associate-/l* 81.1%

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

   7. Applied egg-rr 81.1%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-17} \lor \neg \left(z \leq 2.4 \cdot 10^{-21}\right):\\ \;\;\;\;x\_m \cdot \left(-1 + \frac{1 + y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(1 + y\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= z -7.5e-17) (not (<= z 2.4e-21)))
    (* x_m (+ -1.0 (/ (+ 1.0 y) z)))
    (/ (* 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 ((z <= -7.5e-17) || !(z <= 2.4e-21)) {
		tmp = x_m * (-1.0 + ((1.0 + y) / z));
	} 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 ((z <= (-7.5d-17)) .or. (.not. (z <= 2.4d-21))) then
        tmp = x_m * ((-1.0d0) + ((1.0d0 + y) / z))
    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 ((z <= -7.5e-17) || !(z <= 2.4e-21)) {
		tmp = x_m * (-1.0 + ((1.0 + y) / z));
	} 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 (z <= -7.5e-17) or not (z <= 2.4e-21):
		tmp = x_m * (-1.0 + ((1.0 + y) / z))
	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 ((z <= -7.5e-17) || !(z <= 2.4e-21))
		tmp = Float64(x_m * Float64(-1.0 + Float64(Float64(1.0 + y) / z)));
	else
		tmp = Float64(Float64(x_m * Float64(1.0 + 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 ((z <= -7.5e-17) || ~((z <= 2.4e-21)))
		tmp = x_m * (-1.0 + ((1.0 + y) / z));
	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[Or[LessEqual[z, -7.5e-17], N[Not[LessEqual[z, 2.4e-21]], $MachinePrecision]], N[(x$95$m * N[(-1.0 + N[(N[(1.0 + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -7.49999999999999984e-17 or 2.3999999999999999e-21 < z

   1. Initial program 79.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 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. +-commutative 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   if -7.49999999999999984e-17 < z < 2.3999999999999999e-21

   1. Initial program 99.9%

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

      1. associate-/l* 88.8%

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

      2. +-commutative 88.8%

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

      3. associate-+r- 88.8%

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

      4. div-sub 88.8%

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

      5. *-inverses 88.8%

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

      6. sub-neg 88.8%

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

      7. +-commutative 88.8%

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

      8. metadata-eval 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in z around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-17} \lor \neg \left(z \leq 2.4 \cdot 10^{-21}\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{1 + y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+23}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(y - z\right)\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y}{z} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -6e+23)
    (* x_m (+ (/ y z) -1.0))
    (if (<= z 8.5e+15)
      (* (+ 1.0 (- y z)) (/ x_m 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 <= -6e+23) {
		tmp = x_m * ((y / z) + -1.0);
	} else if (z <= 8.5e+15) {
		tmp = (1.0 + (y - z)) * (x_m / 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 <= (-6d+23)) then
        tmp = x_m * ((y / z) + (-1.0d0))
    else if (z <= 8.5d+15) then
        tmp = (1.0d0 + (y - z)) * (x_m / 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 <= -6e+23) {
		tmp = x_m * ((y / z) + -1.0);
	} else if (z <= 8.5e+15) {
		tmp = (1.0 + (y - z)) * (x_m / 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 <= -6e+23:
		tmp = x_m * ((y / z) + -1.0)
	elif z <= 8.5e+15:
		tmp = (1.0 + (y - z)) * (x_m / 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 <= -6e+23)
		tmp = Float64(x_m * Float64(Float64(y / z) + -1.0));
	elseif (z <= 8.5e+15)
		tmp = Float64(Float64(1.0 + Float64(y - z)) * Float64(x_m / 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 <= -6e+23)
		tmp = x_m * ((y / z) + -1.0);
	elseif (z <= 8.5e+15)
		tmp = (1.0 + (y - z)) * (x_m / 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, -6e+23], N[(x$95$m * N[(N[(y / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e+15], N[(N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $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 < -6.0000000000000002e23

   1. Initial program 70.4%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+r- 100.0%

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

      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. +-commutative 100.0%

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

      8. metadata-eval 100.0%

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

   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 100.0%

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

   if -6.0000000000000002e23 < z < 8.5e15

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

   if 8.5e15 < z

   1. Initial program 82.0%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. +-commutative 99.9%

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

      8. metadata-eval 99.9%

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

   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 99.9%

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

      1. distribute-rgt-in 100.0%

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

      2. neg-mul-1 100.0%

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

      3. unsub-neg 100.0%

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

      4. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

Alternative 5: 65.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+42}:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-14}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -4.9e+42)
    (- x_m)
    (if (<= z -8.5e-14) (* x_m (/ y z)) (if (<= z 1.0) (/ 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 (z <= -4.9e+42) {
		tmp = -x_m;
	} else if (z <= -8.5e-14) {
		tmp = x_m * (y / z);
	} else if (z <= 1.0) {
		tmp = x_m / 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 <= (-4.9d+42)) then
        tmp = -x_m
    else if (z <= (-8.5d-14)) then
        tmp = x_m * (y / z)
    else if (z <= 1.0d0) then
        tmp = x_m / 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 <= -4.9e+42) {
		tmp = -x_m;
	} else if (z <= -8.5e-14) {
		tmp = x_m * (y / z);
	} else if (z <= 1.0) {
		tmp = x_m / 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 <= -4.9e+42:
		tmp = -x_m
	elif z <= -8.5e-14:
		tmp = x_m * (y / z)
	elif z <= 1.0:
		tmp = x_m / 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 <= -4.9e+42)
		tmp = Float64(-x_m);
	elseif (z <= -8.5e-14)
		tmp = Float64(x_m * Float64(y / z));
	elseif (z <= 1.0)
		tmp = Float64(x_m / 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 <= -4.9e+42)
		tmp = -x_m;
	elseif (z <= -8.5e-14)
		tmp = x_m * (y / z);
	elseif (z <= 1.0)
		tmp = x_m / 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, -4.9e+42], (-x$95$m), If[LessEqual[z, -8.5e-14], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(x$95$m / z), $MachinePrecision], (-x$95$m)]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -4.9000000000000002e42 or 1 < z

   1. Initial program 77.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. +-commutative 99.9%

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

      8. metadata-eval 99.9%

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

   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 77.0%

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

      1. neg-mul-1 77.0%

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

   7. Simplified 77.0%

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

   if -4.9000000000000002e42 < z < -8.50000000000000038e-14

   1. Initial program 99.8%

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

      1. associate-/l* 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+r- 99.4%

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

      4. div-sub 99.4%

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

      5. *-inverses 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 87.9%

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

      1. associate-/l* 87.5%

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

   7. Simplified 87.5%

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

   if -8.50000000000000038e-14 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 58.9%

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

      1. *-commutative 58.9%

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

      2. sub-neg 58.9%

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

      3. metadata-eval 58.9%

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

      4. distribute-neg-in 58.9%

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

      5. +-commutative 58.9%

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

      6. associate-/l* 58.9%

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

      7. +-commutative 58.9%

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

      8. distribute-neg-in 58.9%

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

      9. metadata-eval 58.9%

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

      10. sub-neg 58.9%

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

   5. Simplified 58.9%

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

   6. Taylor expanded in z around 0 58.3%

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

Alternative 6: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+20} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -2.55e+20) (not (<= y 1.0)))
    (* x_m (+ (/ y z) -1.0))
    (- (/ 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 <= -2.55e+20) || !(y <= 1.0)) {
		tmp = x_m * ((y / z) + -1.0);
	} 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 <= (-2.55d+20)) .or. (.not. (y <= 1.0d0))) then
        tmp = x_m * ((y / z) + (-1.0d0))
    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 <= -2.55e+20) || !(y <= 1.0)) {
		tmp = x_m * ((y / z) + -1.0);
	} 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 <= -2.55e+20) or not (y <= 1.0):
		tmp = x_m * ((y / z) + -1.0)
	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 <= -2.55e+20) || !(y <= 1.0))
		tmp = Float64(x_m * Float64(Float64(y / z) + -1.0));
	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 <= -2.55e+20) || ~((y <= 1.0)))
		tmp = x_m * ((y / z) + -1.0);
	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, -2.55e+20], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x$95$m * N[(N[(y / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -2.55e20 or 1 < y

   1. Initial program 87.5%

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

      1. associate-/l* 88.8%

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

      2. +-commutative 88.8%

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

      3. associate-+r- 88.8%

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

      4. div-sub 88.9%

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

      5. *-inverses 88.9%

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

      6. sub-neg 88.9%

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

      7. +-commutative 88.9%

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

      8. metadata-eval 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around inf 88.9%

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

   if -2.55e20 < y < 1

   1. Initial program 92.7%

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

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r- 99.8%

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

      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. +-commutative 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-lft-in 99.7%

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

      2. clear-num 99.7%

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

      3. un-div-inv 100.0%

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

      4. *-commutative 100.0%

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

      5. mul-1-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 98.5%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+20} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+40}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x\_m \cdot \left(1 + y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y}{z} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -1.4e+40)
    (* x_m (+ (/ y z) -1.0))
    (if (<= z 1.0) (/ (* x_m (+ 1.0 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 <= -1.4e+40) {
		tmp = x_m * ((y / z) + -1.0);
	} else if (z <= 1.0) {
		tmp = (x_m * (1.0 + 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 <= (-1.4d+40)) then
        tmp = x_m * ((y / z) + (-1.0d0))
    else if (z <= 1.0d0) then
        tmp = (x_m * (1.0d0 + 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 <= -1.4e+40) {
		tmp = x_m * ((y / z) + -1.0);
	} else if (z <= 1.0) {
		tmp = (x_m * (1.0 + 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 <= -1.4e+40:
		tmp = x_m * ((y / z) + -1.0)
	elif z <= 1.0:
		tmp = (x_m * (1.0 + 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 <= -1.4e+40)
		tmp = Float64(x_m * Float64(Float64(y / z) + -1.0));
	elseif (z <= 1.0)
		tmp = Float64(Float64(x_m * Float64(1.0 + 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 <= -1.4e+40)
		tmp = x_m * ((y / z) + -1.0);
	elseif (z <= 1.0)
		tmp = (x_m * (1.0 + 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, -1.4e+40], N[(x$95$m * N[(N[(y / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(x$95$m * N[(1.0 + 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 < -1.4000000000000001e40

   1. Initial program 69.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+r- 100.0%

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

      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. +-commutative 100.0%

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

      8. metadata-eval 100.0%

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

   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 100.0%

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

   if -1.4000000000000001e40 < z < 1

   1. Initial program 99.9%

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

      1. associate-/l* 89.9%

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

      2. +-commutative 89.9%

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

      3. associate-+r- 89.9%

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

      4. div-sub 89.9%

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

      5. *-inverses 89.9%

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

      6. sub-neg 89.9%

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

      7. +-commutative 89.9%

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

      8. metadata-eval 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in z around 0 98.8%

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

   if 1 < z

   1. Initial program 82.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. +-commutative 99.9%

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

      8. metadata-eval 99.9%

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

   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 99.4%

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

      1. distribute-rgt-in 99.4%

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

      2. neg-mul-1 99.4%

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

      3. unsub-neg 99.4%

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

      4. *-commutative 99.4%

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

   7. Applied egg-rr 99.4%

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

Alternative 8: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+20}:\\ \;\;\;\;x\_m \cdot \frac{y}{z} - x\_m\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} + -1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -2.55e+20)
    (- (* x_m (/ y z)) x_m)
    (if (<= y 1.0) (- (/ x_m z) x_m) (* x_m (+ (/ y 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 (y <= -2.55e+20) {
		tmp = (x_m * (y / z)) - x_m;
	} else if (y <= 1.0) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = x_m * ((y / 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 (y <= (-2.55d+20)) then
        tmp = (x_m * (y / z)) - x_m
    else if (y <= 1.0d0) then
        tmp = (x_m / z) - x_m
    else
        tmp = x_m * ((y / 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 (y <= -2.55e+20) {
		tmp = (x_m * (y / z)) - x_m;
	} else if (y <= 1.0) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = x_m * ((y / 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 y <= -2.55e+20:
		tmp = (x_m * (y / z)) - x_m
	elif y <= 1.0:
		tmp = (x_m / z) - x_m
	else:
		tmp = x_m * ((y / 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 (y <= -2.55e+20)
		tmp = Float64(Float64(x_m * Float64(y / z)) - x_m);
	elseif (y <= 1.0)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(x_m * Float64(Float64(y / 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 (y <= -2.55e+20)
		tmp = (x_m * (y / z)) - x_m;
	elseif (y <= 1.0)
		tmp = (x_m / z) - x_m;
	else
		tmp = x_m * ((y / 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[y, -2.55e+20], N[(N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision] - x$95$m), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(x$95$m * N[(N[(y / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -2.55e20

   1. Initial program 87.8%

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

      1. associate-/l* 86.4%

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

      2. +-commutative 86.4%

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

      3. associate-+r- 86.4%

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

      4. div-sub 86.4%

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

      5. *-inverses 86.4%

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

      6. sub-neg 86.4%

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

      7. +-commutative 86.4%

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

      8. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in y around inf 86.4%

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

      1. distribute-rgt-in 86.4%

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

      2. neg-mul-1 86.4%

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

      3. unsub-neg 86.4%

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

      4. *-commutative 86.4%

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

   7. Applied egg-rr 86.4%

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

   if -2.55e20 < y < 1

   1. Initial program 92.7%

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

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r- 99.8%

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

      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. +-commutative 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-lft-in 99.7%

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

      2. clear-num 99.7%

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

      3. un-div-inv 100.0%

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

      4. *-commutative 100.0%

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

      5. mul-1-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 98.5%

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

   if 1 < y

   1. Initial program 87.1%

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

      1. associate-/l* 91.8%

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

      2. +-commutative 91.8%

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

      3. associate-+r- 91.8%

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

      4. div-sub 91.8%

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

      5. *-inverses 91.8%

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

      6. sub-neg 91.8%

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

      7. +-commutative 91.8%

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

      8. metadata-eval 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in y around inf 91.8%

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

Alternative 9: 84.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+55} \lor \neg \left(y \leq 6 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -8.6e+55) (not (<= y 6e+45)))
    (/ (* x_m 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 <= -8.6e+55) || !(y <= 6e+45)) {
		tmp = (x_m * 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 <= (-8.6d+55)) .or. (.not. (y <= 6d+45))) then
        tmp = (x_m * 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 <= -8.6e+55) || !(y <= 6e+45)) {
		tmp = (x_m * 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 <= -8.6e+55) or not (y <= 6e+45):
		tmp = (x_m * 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 <= -8.6e+55) || !(y <= 6e+45))
		tmp = Float64(Float64(x_m * 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 <= -8.6e+55) || ~((y <= 6e+45)))
		tmp = (x_m * 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, -8.6e+55], N[Not[LessEqual[y, 6e+45]], $MachinePrecision]], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -8.5999999999999998e55 or 6.00000000000000021e45 < y

   1. Initial program 89.1%

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

      1. associate-/l* 88.2%

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

      2. +-commutative 88.2%

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

      3. associate-+r- 88.2%

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

      4. div-sub 88.2%

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

      5. *-inverses 88.2%

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

      6. sub-neg 88.2%

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

      7. +-commutative 88.2%

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

      8. metadata-eval 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in y around inf 76.3%

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

   if -8.5999999999999998e55 < y < 6.00000000000000021e45

   1. Initial program 90.8%

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

      1. associate-/l* 99.1%

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

      2. +-commutative 99.1%

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

      3. associate-+r- 99.1%

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

      4. div-sub 99.1%

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

      5. *-inverses 99.1%

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

      6. sub-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. distribute-lft-in 99.1%

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

      2. clear-num 99.1%

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

      3. un-div-inv 99.8%

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

      4. *-commutative 99.8%

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

      5. mul-1-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0 95.2%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+55} \lor \neg \left(y \leq 6 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+55} \lor \neg \left(y \leq 2.5 \cdot 10^{+44}\right):\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -7e+55) (not (<= y 2.5e+44)))
    (* 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 <= -7e+55) || !(y <= 2.5e+44)) {
		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 <= (-7d+55)) .or. (.not. (y <= 2.5d+44))) 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 <= -7e+55) || !(y <= 2.5e+44)) {
		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 <= -7e+55) or not (y <= 2.5e+44):
		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 <= -7e+55) || !(y <= 2.5e+44))
		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 <= -7e+55) || ~((y <= 2.5e+44)))
		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, -7e+55], N[Not[LessEqual[y, 2.5e+44]], $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 < -7.00000000000000021e55 or 2.4999999999999998e44 < y

   1. Initial program 89.1%

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

      1. associate-/l* 88.2%

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

      2. +-commutative 88.2%

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

      3. associate-+r- 88.2%

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

      4. div-sub 88.2%

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

      5. *-inverses 88.2%

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

      6. sub-neg 88.2%

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

      7. +-commutative 88.2%

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

      8. metadata-eval 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in y around inf 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-/l* 73.4%

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

   7. Applied egg-rr 73.4%

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

   if -7.00000000000000021e55 < y < 2.4999999999999998e44

   1. Initial program 90.8%

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

      1. associate-/l* 99.1%

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

      2. +-commutative 99.1%

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

      3. associate-+r- 99.1%

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

      4. div-sub 99.1%

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

      5. *-inverses 99.1%

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

      6. sub-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. distribute-lft-in 99.1%

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

      2. clear-num 99.1%

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

      3. un-div-inv 99.8%

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

      4. *-commutative 99.8%

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

      5. mul-1-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0 95.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+55} \lor \neg \left(y \leq 2.5 \cdot 10^{+44}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 20000000:\\ \;\;\;\;\frac{x\_m \cdot \left(1 + y\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(y - z\right)\right) \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 20000000.0)
    (- (/ (* x_m (+ 1.0 y)) z) x_m)
    (* (+ 1.0 (- y z)) (/ 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 (x_m <= 20000000.0) {
		tmp = ((x_m * (1.0 + y)) / z) - x_m;
	} else {
		tmp = (1.0 + (y - z)) * (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 (x_m <= 20000000.0d0) then
        tmp = ((x_m * (1.0d0 + y)) / z) - x_m
    else
        tmp = (1.0d0 + (y - z)) * (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 (x_m <= 20000000.0) {
		tmp = ((x_m * (1.0 + y)) / z) - x_m;
	} else {
		tmp = (1.0 + (y - z)) * (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 x_m <= 20000000.0:
		tmp = ((x_m * (1.0 + y)) / z) - x_m
	else:
		tmp = (1.0 + (y - z)) * (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 (x_m <= 20000000.0)
		tmp = Float64(Float64(Float64(x_m * Float64(1.0 + y)) / z) - x_m);
	else
		tmp = Float64(Float64(1.0 + Float64(y - z)) * 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 (x_m <= 20000000.0)
		tmp = ((x_m * (1.0 + y)) / z) - x_m;
	else
		tmp = (1.0 + (y - z)) * (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[x$95$m, 20000000.0], N[(N[(N[(x$95$m * N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e7

   1. Initial program 93.3%

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

      1. associate-/l* 92.6%

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

      2. +-commutative 92.6%

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

      3. associate-+r- 92.6%

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

      4. div-sub 92.7%

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

      5. *-inverses 92.7%

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

      6. sub-neg 92.7%

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

      7. +-commutative 92.7%

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

      8. metadata-eval 92.7%

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

   3. Simplified 92.7%

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

      1. distribute-lft-in 92.7%

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

      2. clear-num 92.6%

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

      3. un-div-inv 94.0%

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

      4. *-commutative 94.0%

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

      5. mul-1-neg 94.0%

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

   6. Applied egg-rr 94.0%

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

   7. Taylor expanded in z around inf 97.6%

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

   if 2e7 < x

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

      2. associate-/l* 99.9%

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

      3. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

Alternative 12: 64.5% 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 #s(literal 1 binary64) 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 78.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. +-commutative 99.9%

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

      8. metadata-eval 99.9%

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

   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 73.8%

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

      1. neg-mul-1 73.8%

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

   7. Simplified 73.8%

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

   if -1 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 58.4%

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

      1. *-commutative 58.4%

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

      2. sub-neg 58.4%

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

      3. metadata-eval 58.4%

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

      4. distribute-neg-in 58.4%

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

      5. +-commutative 58.4%

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

      6. associate-/l* 58.4%

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

      7. +-commutative 58.4%

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

      8. distribute-neg-in 58.4%

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

      9. metadata-eval 58.4%

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

      10. sub-neg 58.4%

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

   5. Simplified 58.4%

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

   6. Taylor expanded in z around 0 57.3%

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

4. Final simplification 64.8%

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

Alternative 13: 38.7% 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 #s(literal 1 binary64) 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 90.1%

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

   1. associate-/l* 94.3%

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

   2. +-commutative 94.3%

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

   3. associate-+r- 94.3%

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

   4. div-sub 94.3%

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

   5. *-inverses 94.3%

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

   6. sub-neg 94.3%

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

   7. +-commutative 94.3%

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

   8. metadata-eval 94.3%

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

3. Simplified 94.3%

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

5. Taylor expanded in z around inf 35.1%

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

   1. neg-mul-1 35.1%

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

7. Simplified 35.1%

   \[\leadsto \color{blue}{-x} \]
8. Add Preprocessing

Alternative 14: 3.0% accurate, 9.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 * 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 90.1%

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

   1. associate-/l* 94.3%

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

   2. +-commutative 94.3%

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

   3. associate-+r- 94.3%

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

   4. div-sub 94.3%

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

   5. *-inverses 94.3%

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

   6. sub-neg 94.3%

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

   7. +-commutative 94.3%

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

   8. metadata-eval 94.3%

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

3. Simplified 94.3%

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

5. Taylor expanded in z around inf 35.1%

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

   1. neg-mul-1 35.1%

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

7. Simplified 35.1%

   \[\leadsto \color{blue}{-x} \]
8. Step-by-step derivation

   1. neg-sub0 35.1%

      \[\leadsto \color{blue}{0 - x} \]

   2. sub-neg 35.1%

      \[\leadsto \color{blue}{0 + \left(-x\right)} \]

   3. add-sqr-sqrt 17.4%

      \[\leadsto 0 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}} \]

   4. sqrt-unprod 20.4%

      \[\leadsto 0 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \]

   5. sqr-neg 20.4%

      \[\leadsto 0 + \sqrt{\color{blue}{x \cdot x}} \]

   6. sqrt-unprod 1.6%

      \[\leadsto 0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}} \]

   7. add-sqr-sqrt 3.0%

      \[\leadsto 0 + \color{blue}{x} \]

9. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{0 + x} \]
10. Step-by-step derivation

    1. +-lft-identity 3.0%

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

11. Simplified 3.0%

    \[\leadsto \color{blue}{x} \]
12. Add Preprocessing

Developer Target 1: 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 2024157 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -67870776678359/25000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (+ 1 y) (/ x z)) x) (if (< x 1937054408219773/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x))))

  (/ (* x (+ (- y z) 1.0)) z))