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

Percentage Accurate: 87.8% → 99.4%

Time: 8.1s

Alternatives: 15

Speedup: 1.0×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.8% 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: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{x\_m \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{y + 1}} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.2e-101)
    (/ (* x_m (+ (- y z) 1.0)) z)
    (- (/ x_m (/ z (+ y 1.0))) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.2e-101) {
		tmp = (x_m * ((y - z) + 1.0)) / z;
	} else {
		tmp = (x_m / (z / (y + 1.0))) - x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 2.2d-101) then
        tmp = (x_m * ((y - z) + 1.0d0)) / z
    else
        tmp = (x_m / (z / (y + 1.0d0))) - x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.2e-101) {
		tmp = (x_m * ((y - z) + 1.0)) / z;
	} else {
		tmp = (x_m / (z / (y + 1.0))) - x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 2.2e-101:
		tmp = (x_m * ((y - z) + 1.0)) / z
	else:
		tmp = (x_m / (z / (y + 1.0))) - x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 2.2e-101)
		tmp = Float64(Float64(x_m * Float64(Float64(y - z) + 1.0)) / z);
	else
		tmp = Float64(Float64(x_m / Float64(z / Float64(y + 1.0))) - x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 2.2e-101)
		tmp = (x_m * ((y - z) + 1.0)) / z;
	else
		tmp = (x_m / (z / (y + 1.0))) - x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 2.2e-101], N[(N[(x$95$m * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / N[(z / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.1999999999999999e-101

   1. Initial program 89.8%

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

   if 2.1999999999999999e-101 < x

   1. Initial program 85.0%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 99.8%

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

      2. clear-num 99.8%

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

      3. un-div-inv 99.9%

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

      4. *-commutative 99.9%

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

      5. mul-1-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + 1}} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -25:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\_m\\ \end{array} \end{array} \]

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 -25.0)
    (- x_m)
    (if (<= z -7.2e-99) (* 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 <= -25.0) {
		tmp = -x_m;
	} else if (z <= -7.2e-99) {
		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 <= (-25.0d0)) then
        tmp = -x_m
    else if (z <= (-7.2d-99)) 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 <= -25.0) {
		tmp = -x_m;
	} else if (z <= -7.2e-99) {
		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 <= -25.0:
		tmp = -x_m
	elif z <= -7.2e-99:
		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 <= -25.0)
		tmp = Float64(-x_m);
	elseif (z <= -7.2e-99)
		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 <= -25.0)
		tmp = -x_m;
	elseif (z <= -7.2e-99)
		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, -25.0], (-x$95$m), If[LessEqual[z, -7.2e-99], 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 3 regimes

2. if z < -25 or 1 < z

   1. Initial program 75.1%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 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 74.4%

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

      1. neg-mul-1 74.4%

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

   7. Simplified 74.4%

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

   if -25 < z < -7.2000000000000001e-99

   1. Initial program 99.8%

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

      1. associate-/l* 94.8%

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

      2. +-commutative 94.8%

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

      3. associate-+r- 94.8%

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

      4. div-sub 94.8%

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

      5. *-inverses 94.8%

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

      6. sub-neg 94.8%

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

      7. +-commutative 94.8%

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

      8. metadata-eval 94.8%

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

   3. Simplified 94.8%

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

      1. distribute-lft-in 94.8%

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

      2. clear-num 94.8%

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

      3. un-div-inv 95.0%

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

      4. *-commutative 95.0%

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

      5. mul-1-neg 95.0%

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

   6. Applied egg-rr 95.0%

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

   7. Taylor expanded in y around inf 74.7%

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

      1. associate-*l/ 74.5%

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

      2. *-commutative 74.5%

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

   9. Simplified 74.5%

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

   if -7.2000000000000001e-99 < z < 1

   1. Initial program 100.0%

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

      1. associate-/l* 94.2%

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

      2. +-commutative 94.2%

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

      3. associate-+r- 94.2%

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

      4. div-sub 94.2%

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

      5. *-inverses 94.2%

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

      6. sub-neg 94.2%

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

      7. +-commutative 94.2%

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

      8. metadata-eval 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in y around 0 60.4%

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

      1. sub-neg 60.4%

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

      2. metadata-eval 60.4%

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

      3. distribute-rgt-in 60.3%

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

      4. associate-*l/ 60.4%

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

      5. *-lft-identity 60.4%

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

      6. neg-mul-1 60.4%

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

      7. unsub-neg 60.4%

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

   7. Simplified 60.4%

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

   8. Taylor expanded in z around 0 60.1%

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

Alternative 3: 65.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -25:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-92}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\_m\\ \end{array} \end{array} \]

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 -25.0)
    (- x_m)
    (if (<= z -1.22e-92) (* 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 <= -25.0) {
		tmp = -x_m;
	} else if (z <= -1.22e-92) {
		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 <= (-25.0d0)) then
        tmp = -x_m
    else if (z <= (-1.22d-92)) 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 <= -25.0) {
		tmp = -x_m;
	} else if (z <= -1.22e-92) {
		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 <= -25.0:
		tmp = -x_m
	elif z <= -1.22e-92:
		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 <= -25.0)
		tmp = Float64(-x_m);
	elseif (z <= -1.22e-92)
		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 <= -25.0)
		tmp = -x_m;
	elseif (z <= -1.22e-92)
		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, -25.0], (-x$95$m), If[LessEqual[z, -1.22e-92], 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 < -25 or 1 < z

   1. Initial program 75.1%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 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 74.4%

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

      1. neg-mul-1 74.4%

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

   7. Simplified 74.4%

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

   if -25 < z < -1.21999999999999994e-92

   1. Initial program 99.8%

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

      1. associate-/l* 94.8%

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

      2. +-commutative 94.8%

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

      3. associate-+r- 94.8%

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

      4. div-sub 94.8%

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

      5. *-inverses 94.8%

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

      6. sub-neg 94.8%

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

      7. +-commutative 94.8%

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

      8. metadata-eval 94.8%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in y around inf 74.7%

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

      1. associate-/l* 69.7%

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

   7. Simplified 69.7%

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

   if -1.21999999999999994e-92 < z < 1

   1. Initial program 100.0%

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

      1. associate-/l* 94.2%

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

      2. +-commutative 94.2%

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

      3. associate-+r- 94.2%

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

      4. div-sub 94.2%

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

      5. *-inverses 94.2%

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

      6. sub-neg 94.2%

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

      7. +-commutative 94.2%

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

      8. metadata-eval 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in y around 0 60.4%

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

      1. sub-neg 60.4%

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

      2. metadata-eval 60.4%

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

      3. distribute-rgt-in 60.3%

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

      4. associate-*l/ 60.4%

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

      5. *-lft-identity 60.4%

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

      6. neg-mul-1 60.4%

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

      7. unsub-neg 60.4%

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

   7. Simplified 60.4%

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

   8. Taylor expanded in z around 0 60.1%

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

Alternative 4: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.26\right):\\ \;\;\;\;x\_m \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -1.0) (not (<= y 0.26)))
    (* x_m (+ -1.0 (/ y z)))
    (- (/ x_m z) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.26)) {
		tmp = x_m * (-1.0 + (y / z));
	} else {
		tmp = (x_m / z) - x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 0.26d0))) then
        tmp = x_m * ((-1.0d0) + (y / z))
    else
        tmp = (x_m / z) - x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.26)) {
		tmp = x_m * (-1.0 + (y / z));
	} else {
		tmp = (x_m / z) - x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.26):
		tmp = x_m * (-1.0 + (y / z))
	else:
		tmp = (x_m / z) - x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.26))
		tmp = Float64(x_m * Float64(-1.0 + Float64(y / z)));
	else
		tmp = Float64(Float64(x_m / z) - x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.26)))
		tmp = x_m * (-1.0 + (y / z));
	else
		tmp = (x_m / z) - x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.26]], $MachinePrecision]], N[(x$95$m * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.26000000000000001 < y

   1. Initial program 87.2%

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

      1. associate-/l* 94.1%

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

      2. +-commutative 94.1%

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

      3. associate-+r- 94.1%

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

      4. div-sub 94.1%

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

      5. *-inverses 94.1%

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

      6. sub-neg 94.1%

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

      7. +-commutative 94.1%

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

      8. metadata-eval 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in y around inf 93.4%

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

   if -1 < y < 0.26000000000000001

   1. Initial program 89.4%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. Taylor expanded in y around 0 99.0%

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

      1. sub-neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. distribute-rgt-in 99.0%

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

      4. associate-*l/ 99.2%

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

      5. *-lft-identity 99.2%

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

      6. neg-mul-1 99.2%

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

      7. unsub-neg 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.26\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.05:\\ \;\;\;\;x\_m \cdot \frac{y}{z} - x\_m\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x\_m + x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{y}} - x\_m\\ \end{array} \end{array} \]

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 (<= z -1.05)
    (- (* x_m (/ y z)) x_m)
    (if (<= z 1.0) (/ (+ x_m (* x_m y)) z) (- (/ x_m (/ z 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 (z <= -1.05) {
		tmp = (x_m * (y / z)) - x_m;
	} else if (z <= 1.0) {
		tmp = (x_m + (x_m * y)) / z;
	} else {
		tmp = (x_m / (z / 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 (z <= (-1.05d0)) then
        tmp = (x_m * (y / z)) - x_m
    else if (z <= 1.0d0) then
        tmp = (x_m + (x_m * y)) / z
    else
        tmp = (x_m / (z / 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 (z <= -1.05) {
		tmp = (x_m * (y / z)) - x_m;
	} else if (z <= 1.0) {
		tmp = (x_m + (x_m * y)) / z;
	} else {
		tmp = (x_m / (z / 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 z <= -1.05:
		tmp = (x_m * (y / z)) - x_m
	elif z <= 1.0:
		tmp = (x_m + (x_m * y)) / z
	else:
		tmp = (x_m / (z / 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 (z <= -1.05)
		tmp = Float64(Float64(x_m * Float64(y / z)) - x_m);
	elseif (z <= 1.0)
		tmp = Float64(Float64(x_m + Float64(x_m * y)) / z);
	else
		tmp = Float64(Float64(x_m / Float64(z / 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 (z <= -1.05)
		tmp = (x_m * (y / z)) - x_m;
	elseif (z <= 1.0)
		tmp = (x_m + (x_m * y)) / z;
	else
		tmp = (x_m / (z / 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[z, -1.05], N[(N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision] - x$95$m), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(x$95$m + N[(x$95$m * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / N[(z / y), $MachinePrecision]), $MachinePrecision] - x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.05000000000000004

   1. Initial program 74.1%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 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 100.0%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

      4. *-commutative 99.9%

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

      5. mul-1-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf 96.8%

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

      1. unsub-neg 96.8%

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

      2. div-inv 96.7%

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

      3. clear-num 96.8%

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

   9. Applied egg-rr 96.8%

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

   if -1.05000000000000004 < z < 1

   1. Initial program 99.9%

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

      1. distribute-lft-in 99.9%

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

      2. fma-define 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.3%

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

   if 1 < z

   1. Initial program 76.2%

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

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

   7. Taylor expanded in y around inf 97.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 0.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -1.0)
    (- (* x_m (/ y z)) x_m)
    (if (<= z 1.0) (/ (+ x_m (* x_m 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 <= -1.0) {
		tmp = (x_m * (y / z)) - x_m;
	} else if (z <= 1.0) {
		tmp = (x_m + (x_m * 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 <= (-1.0d0)) then
        tmp = (x_m * (y / z)) - x_m
    else if (z <= 1.0d0) then
        tmp = (x_m + (x_m * 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 <= -1.0) {
		tmp = (x_m * (y / z)) - x_m;
	} else if (z <= 1.0) {
		tmp = (x_m + (x_m * 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 <= -1.0:
		tmp = (x_m * (y / z)) - x_m
	elif z <= 1.0:
		tmp = (x_m + (x_m * 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 <= -1.0)
		tmp = Float64(Float64(x_m * Float64(y / z)) - x_m);
	elseif (z <= 1.0)
		tmp = Float64(Float64(x_m + Float64(x_m * y)) / z);
	else
		tmp = Float64(x_m * Float64(-1.0 + Float64(y / z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (x_m * (y / z)) - x_m;
	elseif (z <= 1.0)
		tmp = (x_m + (x_m * 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[LessEqual[z, -1.0], N[(N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision] - x$95$m), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(x$95$m + N[(x$95$m * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 74.1%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 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 100.0%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

      4. *-commutative 99.9%

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

      5. mul-1-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf 96.8%

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

      1. unsub-neg 96.8%

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

      2. div-inv 96.7%

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

      3. clear-num 96.8%

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

   9. Applied egg-rr 96.8%

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

   if -1 < z < 1

   1. Initial program 99.9%

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

      1. distribute-lft-in 99.9%

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

      2. fma-define 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.3%

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

   if 1 < z

   1. Initial program 76.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 97.0%

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

4. Final simplification 98.2%

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

Alternative 7: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\_m \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{elif}\;y \leq 0.26:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y}{z} - x\_m\\ \end{array} \end{array} \]

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 (<= y -1.0)
    (* x_m (+ -1.0 (/ y z)))
    (if (<= y 0.26) (- (/ x_m z) x_m) (- (* 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 (y <= -1.0) {
		tmp = x_m * (-1.0 + (y / z));
	} else if (y <= 0.26) {
		tmp = (x_m / z) - x_m;
	} 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 (y <= (-1.0d0)) then
        tmp = x_m * ((-1.0d0) + (y / z))
    else if (y <= 0.26d0) then
        tmp = (x_m / z) - x_m
    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 (y <= -1.0) {
		tmp = x_m * (-1.0 + (y / z));
	} else if (y <= 0.26) {
		tmp = (x_m / z) - x_m;
	} 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 y <= -1.0:
		tmp = x_m * (-1.0 + (y / z))
	elif y <= 0.26:
		tmp = (x_m / z) - x_m
	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 (y <= -1.0)
		tmp = Float64(x_m * Float64(-1.0 + Float64(y / z)));
	elseif (y <= 0.26)
		tmp = Float64(Float64(x_m / z) - x_m);
	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 (y <= -1.0)
		tmp = x_m * (-1.0 + (y / z));
	elseif (y <= 0.26)
		tmp = (x_m / z) - x_m;
	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[y, -1.0], N[(x$95$m * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.26], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $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 y < -1

   1. Initial program 87.3%

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

      1. associate-/l* 96.9%

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

      2. +-commutative 96.9%

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

      3. associate-+r- 96.9%

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

      4. div-sub 96.9%

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

      5. *-inverses 96.9%

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

      6. sub-neg 96.9%

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

      7. +-commutative 96.9%

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

      8. metadata-eval 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in y around inf 96.4%

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

   if -1 < y < 0.26000000000000001

   1. Initial program 89.4%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. Taylor expanded in y around 0 99.0%

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

      1. sub-neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. distribute-rgt-in 99.0%

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

      4. associate-*l/ 99.2%

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

      5. *-lft-identity 99.2%

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

      6. neg-mul-1 99.2%

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

      7. unsub-neg 99.2%

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

   7. Simplified 99.2%

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

   if 0.26000000000000001 < y

   1. Initial program 87.0%

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

      1. associate-/l* 91.2%

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

      2. +-commutative 91.2%

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

      3. associate-+r- 91.2%

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

      4. div-sub 91.2%

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

      5. *-inverses 91.2%

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

      6. sub-neg 91.2%

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

      7. +-commutative 91.2%

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

      8. metadata-eval 91.2%

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

   3. Simplified 91.2%

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

      1. distribute-lft-in 91.2%

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

      2. clear-num 91.2%

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

      3. un-div-inv 91.2%

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

      4. *-commutative 91.2%

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

      5. mul-1-neg 91.2%

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

   6. Applied egg-rr 91.2%

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

   7. Taylor expanded in y around inf 90.4%

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

      1. unsub-neg 90.4%

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

      2. div-inv 90.3%

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

      3. clear-num 90.4%

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

   9. Applied egg-rr 90.4%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{elif}\;y \leq 0.26:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+36} \lor \neg \left(y \leq 1.4 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

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 -7e+36) (not (<= y 1.4e+50)))
    (/ (* 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 <= -7e+36) || !(y <= 1.4e+50)) {
		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 <= (-7d+36)) .or. (.not. (y <= 1.4d+50))) 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 <= -7e+36) || !(y <= 1.4e+50)) {
		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 <= -7e+36) or not (y <= 1.4e+50):
		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 <= -7e+36) || !(y <= 1.4e+50))
		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 <= -7e+36) || ~((y <= 1.4e+50)))
		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, -7e+36], N[Not[LessEqual[y, 1.4e+50]], $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 < -6.9999999999999996e36 or 1.3999999999999999e50 < y

   1. Initial program 90.3%

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

      1. associate-/l* 92.7%

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

      2. +-commutative 92.7%

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

      3. associate-+r- 92.7%

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

      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. Taylor expanded in y around inf 75.1%

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

   if -6.9999999999999996e36 < y < 1.3999999999999999e50

   1. Initial program 86.9%

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

   5. Taylor expanded in y around 0 93.5%

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

      1. sub-neg 93.5%

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

      2. metadata-eval 93.5%

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

      3. distribute-rgt-in 93.5%

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

      4. associate-*l/ 93.6%

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

      5. *-lft-identity 93.6%

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

      6. neg-mul-1 93.6%

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

      7. unsub-neg 93.6%

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

   7. Simplified 93.6%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+36} \lor \neg \left(y \leq 1.4 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -4.8e+36)
    (/ x_m (/ z y))
    (if (<= y 7.5e+51) (- (/ x_m z) x_m) (* y (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -4.8e+36) {
		tmp = x_m / (z / y);
	} else if (y <= 7.5e+51) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = y * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.8d+36)) then
        tmp = x_m / (z / y)
    else if (y <= 7.5d+51) then
        tmp = (x_m / z) - x_m
    else
        tmp = y * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -4.8e+36) {
		tmp = x_m / (z / y);
	} else if (y <= 7.5e+51) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = y * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -4.8e+36:
		tmp = x_m / (z / y)
	elif y <= 7.5e+51:
		tmp = (x_m / z) - x_m
	else:
		tmp = y * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -4.8e+36)
		tmp = Float64(x_m / Float64(z / y));
	elseif (y <= 7.5e+51)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(y * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -4.8e+36)
		tmp = x_m / (z / y);
	elseif (y <= 7.5e+51)
		tmp = (x_m / z) - x_m;
	else
		tmp = y * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -4.8e+36], N[(x$95$m / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+51], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -4.79999999999999985e36

   1. Initial program 90.1%

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

      1. associate-/l* 96.5%

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

      2. +-commutative 96.5%

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

      3. associate-+r- 96.5%

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

      4. div-sub 96.5%

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

      5. *-inverses 96.5%

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

      6. sub-neg 96.5%

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

      7. +-commutative 96.5%

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

      8. metadata-eval 96.5%

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

   3. Simplified 96.5%

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

      1. distribute-lft-in 96.5%

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

      2. clear-num 96.4%

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

      3. un-div-inv 96.5%

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

      4. *-commutative 96.5%

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

      5. mul-1-neg 96.5%

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

   6. Applied egg-rr 96.5%

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

   7. Taylor expanded in y around inf 96.5%

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

   8. Taylor expanded in z around 0 70.7%

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

      1. associate-*l/ 69.1%

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

      2. associate-/r/ 69.1%

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

   10. Simplified 69.1%

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

   if -4.79999999999999985e36 < y < 7.4999999999999999e51

   1. Initial program 86.9%

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

   5. Taylor expanded in y around 0 93.5%

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

      1. sub-neg 93.5%

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

      2. metadata-eval 93.5%

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

      3. distribute-rgt-in 93.5%

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

      4. associate-*l/ 93.6%

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

      5. *-lft-identity 93.6%

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

      6. neg-mul-1 93.6%

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

      7. unsub-neg 93.6%

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

   7. Simplified 93.6%

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

   if 7.4999999999999999e51 < y

   1. Initial program 90.4%

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

      1. associate-/l* 88.4%

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

      2. +-commutative 88.4%

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

      3. associate-+r- 88.4%

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

      4. div-sub 88.4%

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

      5. *-inverses 88.4%

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

      6. sub-neg 88.4%

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

      7. +-commutative 88.4%

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

      8. metadata-eval 88.4%

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

   3. Simplified 88.4%

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

      1. distribute-lft-in 88.4%

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

      2. clear-num 88.3%

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

      3. un-div-inv 88.4%

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

      4. *-commutative 88.4%

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

      5. mul-1-neg 88.4%

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

   6. Applied egg-rr 88.4%

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

   7. Taylor expanded in y around inf 80.2%

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

      1. associate-*l/ 76.7%

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

      2. *-commutative 76.7%

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

   9. Simplified 76.7%

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

Alternative 10: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+37}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -2.9e+37)
    (* x_m (/ y z))
    (if (<= y 5.5e+49) (- (/ x_m z) x_m) (* y (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -2.9e+37) {
		tmp = x_m * (y / z);
	} else if (y <= 5.5e+49) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = y * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.9d+37)) then
        tmp = x_m * (y / z)
    else if (y <= 5.5d+49) then
        tmp = (x_m / z) - x_m
    else
        tmp = y * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -2.9e+37) {
		tmp = x_m * (y / z);
	} else if (y <= 5.5e+49) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = y * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -2.9e+37:
		tmp = x_m * (y / z)
	elif y <= 5.5e+49:
		tmp = (x_m / z) - x_m
	else:
		tmp = y * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -2.9e+37)
		tmp = Float64(x_m * Float64(y / z));
	elseif (y <= 5.5e+49)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(y * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -2.9e+37)
		tmp = x_m * (y / z);
	elseif (y <= 5.5e+49)
		tmp = (x_m / z) - x_m;
	else
		tmp = y * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -2.9e+37], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e+49], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -2.89999999999999978e37

   1. Initial program 90.1%

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

      1. associate-/l* 96.5%

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

      2. +-commutative 96.5%

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

      3. associate-+r- 96.5%

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

      4. div-sub 96.5%

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

      5. *-inverses 96.5%

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

      6. sub-neg 96.5%

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

      7. +-commutative 96.5%

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

      8. metadata-eval 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in y around inf 70.7%

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

      1. associate-/l* 69.1%

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

   7. Simplified 69.1%

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

   if -2.89999999999999978e37 < y < 5.50000000000000042e49

   1. Initial program 86.9%

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

   5. Taylor expanded in y around 0 93.5%

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

      1. sub-neg 93.5%

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

      2. metadata-eval 93.5%

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

      3. distribute-rgt-in 93.5%

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

      4. associate-*l/ 93.6%

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

      5. *-lft-identity 93.6%

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

      6. neg-mul-1 93.6%

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

      7. unsub-neg 93.6%

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

   7. Simplified 93.6%

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

   if 5.50000000000000042e49 < y

   1. Initial program 90.4%

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

      1. associate-/l* 88.4%

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

      2. +-commutative 88.4%

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

      3. associate-+r- 88.4%

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

      4. div-sub 88.4%

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

      5. *-inverses 88.4%

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

      6. sub-neg 88.4%

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

      7. +-commutative 88.4%

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

      8. metadata-eval 88.4%

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

   3. Simplified 88.4%

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

      1. distribute-lft-in 88.4%

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

      2. clear-num 88.3%

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

      3. un-div-inv 88.4%

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

      4. *-commutative 88.4%

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

      5. mul-1-neg 88.4%

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

   6. Applied egg-rr 88.4%

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

   7. Taylor expanded in y around inf 80.2%

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

      1. associate-*l/ 76.7%

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

      2. *-commutative 76.7%

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

   9. Simplified 76.7%

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

Alternative 11: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x\_m \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y + 1}{z} + -1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2e-5)
    (/ (* x_m (+ (- y z) 1.0)) z)
    (* x_m (+ (/ (+ y 1.0) z) -1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2e-5) {
		tmp = (x_m * ((y - z) + 1.0)) / z;
	} else {
		tmp = x_m * (((y + 1.0) / z) + -1.0);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 2d-5) then
        tmp = (x_m * ((y - z) + 1.0d0)) / z
    else
        tmp = x_m * (((y + 1.0d0) / z) + (-1.0d0))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2e-5) {
		tmp = (x_m * ((y - z) + 1.0)) / z;
	} else {
		tmp = x_m * (((y + 1.0) / z) + -1.0);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 2e-5:
		tmp = (x_m * ((y - z) + 1.0)) / z
	else:
		tmp = x_m * (((y + 1.0) / z) + -1.0)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 2e-5)
		tmp = Float64(Float64(x_m * Float64(Float64(y - z) + 1.0)) / z);
	else
		tmp = Float64(x_m * Float64(Float64(Float64(y + 1.0) / z) + -1.0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 2e-5)
		tmp = (x_m * ((y - z) + 1.0)) / z;
	else
		tmp = x_m * (((y + 1.0) / z) + -1.0);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 2e-5], N[(N[(x$95$m * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * N[(N[(N[(y + 1.0), $MachinePrecision] / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000016e-5

   1. Initial program 90.8%

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

   if 2.00000000000000016e-5 < 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
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 65.5% 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 75.1%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+r- 99.9%

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

      4. div-sub 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 74.4%

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

      1. neg-mul-1 74.4%

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

   7. Simplified 74.4%

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

   if -1 < z < 1

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. 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 y around 0 55.6%

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

      1. sub-neg 55.6%

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

      2. metadata-eval 55.6%

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

      3. distribute-rgt-in 55.6%

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

      4. associate-*l/ 55.7%

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

      5. *-lft-identity 55.7%

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

      6. neg-mul-1 55.7%

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

      7. unsub-neg 55.7%

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

   7. Simplified 55.7%

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

   8. Taylor expanded in z around 0 55.1%

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

4. Final simplification 64.2%

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

Alternative 13: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

   1. associate-/l* 96.9%

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

   2. +-commutative 96.9%

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

   3. associate-+r- 96.9%

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

   4. div-sub 96.9%

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

   5. *-inverses 96.9%

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

   6. sub-neg 96.9%

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

   7. +-commutative 96.9%

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

   8. metadata-eval 96.9%

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

3. Simplified 96.9%

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

Alternative 14: 38.8% 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 88.3%

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

   1. associate-/l* 96.9%

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

   2. +-commutative 96.9%

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

   3. associate-+r- 96.9%

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

   4. div-sub 96.9%

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

   5. *-inverses 96.9%

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

   6. sub-neg 96.9%

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

   7. +-commutative 96.9%

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

   8. metadata-eval 96.9%

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

3. Simplified 96.9%

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

5. Taylor expanded in z around inf 36.5%

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

   1. neg-mul-1 36.5%

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

7. Simplified 36.5%

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

Alternative 15: 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 88.3%

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

   1. associate-/l* 96.9%

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

   2. +-commutative 96.9%

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

   3. associate-+r- 96.9%

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

   4. div-sub 96.9%

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

   5. *-inverses 96.9%

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

   6. sub-neg 96.9%

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

   7. +-commutative 96.9%

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

   8. metadata-eval 96.9%

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

3. Simplified 96.9%

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

5. Taylor expanded in z around inf 36.5%

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

   1. neg-mul-1 36.5%

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

7. Simplified 36.5%

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

   1. neg-sub0 36.5%

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

   2. sub-neg 36.5%

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

   3. add-sqr-sqrt 19.9%

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

   4. sqrt-unprod 18.2%

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

   5. sqr-neg 18.2%

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

   6. sqrt-unprod 1.5%

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

   7. add-sqr-sqrt 3.2%

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

9. Applied egg-rr 3.2%

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

    1. +-lft-identity 3.2%

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

11. Simplified 3.2%

    \[\leadsto \color{blue}{x} \]
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 2024129 
(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))