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

Percentage Accurate: 87.7% → 99.9%

Time: 8.6s

Alternatives: 15

Speedup: 1.0×

• 20.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x\_m \cdot \left(1 + y\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{1 + \left(y - z\right)}}\\ \end{array} \end{array} \]

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 (<= x_m 4e-14)
    (- (/ (* x_m (+ 1.0 y)) z) x_m)
    (/ x_m (/ z (+ 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 (x_m <= 4e-14) {
		tmp = ((x_m * (1.0 + y)) / z) - x_m;
	} else {
		tmp = x_m / (z / (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 (x_m <= 4d-14) then
        tmp = ((x_m * (1.0d0 + y)) / z) - x_m
    else
        tmp = x_m / (z / (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 (x_m <= 4e-14) {
		tmp = ((x_m * (1.0 + y)) / z) - x_m;
	} else {
		tmp = x_m / (z / (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 x_m <= 4e-14:
		tmp = ((x_m * (1.0 + y)) / z) - x_m
	else:
		tmp = x_m / (z / (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 (x_m <= 4e-14)
		tmp = Float64(Float64(Float64(x_m * Float64(1.0 + y)) / z) - x_m);
	else
		tmp = Float64(x_m / Float64(z / 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 (x_m <= 4e-14)
		tmp = ((x_m * (1.0 + y)) / z) - x_m;
	else
		tmp = x_m / (z / (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[x$95$m, 4e-14], N[(N[(N[(x$95$m * N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(x$95$m / N[(z / N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4e-14

   1. Initial program 89.1%

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

      1. associate-/l* 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in z around inf 96.3%

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

   if 4e-14 < x

   1. Initial program 75.4%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+75}:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-47}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+107}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y z))))
   (*
    x_s
    (if (<= z -3.9e+75)
      (- x_m)
      (if (<= z -6.8e-65)
        t_0
        (if (<= z 1.15e-47) (/ x_m z) (if (<= z 1.45e+107) t_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 t_0 = x_m * (y / z);
	double tmp;
	if (z <= -3.9e+75) {
		tmp = -x_m;
	} else if (z <= -6.8e-65) {
		tmp = t_0;
	} else if (z <= 1.15e-47) {
		tmp = x_m / z;
	} else if (z <= 1.45e+107) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x_m * (y / z)
    if (z <= (-3.9d+75)) then
        tmp = -x_m
    else if (z <= (-6.8d-65)) then
        tmp = t_0
    else if (z <= 1.15d-47) then
        tmp = x_m / z
    else if (z <= 1.45d+107) then
        tmp = t_0
    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 t_0 = x_m * (y / z);
	double tmp;
	if (z <= -3.9e+75) {
		tmp = -x_m;
	} else if (z <= -6.8e-65) {
		tmp = t_0;
	} else if (z <= 1.15e-47) {
		tmp = x_m / z;
	} else if (z <= 1.45e+107) {
		tmp = t_0;
	} 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):
	t_0 = x_m * (y / z)
	tmp = 0
	if z <= -3.9e+75:
		tmp = -x_m
	elif z <= -6.8e-65:
		tmp = t_0
	elif z <= 1.15e-47:
		tmp = x_m / z
	elif z <= 1.45e+107:
		tmp = t_0
	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)
	t_0 = Float64(x_m * Float64(y / z))
	tmp = 0.0
	if (z <= -3.9e+75)
		tmp = Float64(-x_m);
	elseif (z <= -6.8e-65)
		tmp = t_0;
	elseif (z <= 1.15e-47)
		tmp = Float64(x_m / z);
	elseif (z <= 1.45e+107)
		tmp = t_0;
	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)
	t_0 = x_m * (y / z);
	tmp = 0.0;
	if (z <= -3.9e+75)
		tmp = -x_m;
	elseif (z <= -6.8e-65)
		tmp = t_0;
	elseif (z <= 1.15e-47)
		tmp = x_m / z;
	elseif (z <= 1.45e+107)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -3.9e+75], (-x$95$m), If[LessEqual[z, -6.8e-65], t$95$0, If[LessEqual[z, 1.15e-47], N[(x$95$m / z), $MachinePrecision], If[LessEqual[z, 1.45e+107], t$95$0, (-x$95$m)]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -3.90000000000000038e75 or 1.44999999999999994e107 < z

   1. Initial program 64.8%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 81.4%

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

      1. neg-mul-1 81.4%

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

   7. Simplified 81.4%

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

   if -3.90000000000000038e75 < z < -6.79999999999999973e-65 or 1.14999999999999991e-47 < z < 1.44999999999999994e107

   1. Initial program 94.6%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 60.8%

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

      1. associate-/l* 64.1%

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

   7. Simplified 64.1%

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

   if -6.79999999999999973e-65 < z < 1.14999999999999991e-47

   1. Initial program 99.9%

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

      1. associate-/l* 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in z around 0 99.9%

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

      1. associate-/l* 90.5%

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

   7. Simplified 90.5%

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

   8. Taylor expanded in y around 0 57.2%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := 1 + \left(y - z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-47} \lor \neg \left(z \leq 5 \cdot 10^{-13}\right):\\ \;\;\;\;x\_m \cdot \frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (- y z))))
   (*
    x_s
    (if (or (<= z -2.5e-47) (not (<= z 5e-13)))
      (* x_m (/ t_0 z))
      (* t_0 (/ 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 t_0 = 1.0 + (y - z);
	double tmp;
	if ((z <= -2.5e-47) || !(z <= 5e-13)) {
		tmp = x_m * (t_0 / z);
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (y - z)
    if ((z <= (-2.5d-47)) .or. (.not. (z <= 5d-13))) then
        tmp = x_m * (t_0 / z)
    else
        tmp = t_0 * (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 t_0 = 1.0 + (y - z);
	double tmp;
	if ((z <= -2.5e-47) || !(z <= 5e-13)) {
		tmp = x_m * (t_0 / z);
	} else {
		tmp = t_0 * (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):
	t_0 = 1.0 + (y - z)
	tmp = 0
	if (z <= -2.5e-47) or not (z <= 5e-13):
		tmp = x_m * (t_0 / z)
	else:
		tmp = t_0 * (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)
	t_0 = Float64(1.0 + Float64(y - z))
	tmp = 0.0
	if ((z <= -2.5e-47) || !(z <= 5e-13))
		tmp = Float64(x_m * Float64(t_0 / z));
	else
		tmp = Float64(t_0 * 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)
	t_0 = 1.0 + (y - z);
	tmp = 0.0;
	if ((z <= -2.5e-47) || ~((z <= 5e-13)))
		tmp = x_m * (t_0 / z);
	else
		tmp = t_0 * (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_] := Block[{t$95$0 = N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[Or[LessEqual[z, -2.5e-47], N[Not[LessEqual[z, 5e-13]], $MachinePrecision]], N[(x$95$m * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.50000000000000006e-47 or 4.9999999999999999e-13 < z

   1. Initial program 73.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   if -2.50000000000000006e-47 < z < 4.9999999999999999e-13

   1. Initial program 99.9%

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

      1. associate-/l* 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*l/ 99.9%

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

      4. +-commutative 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-47} \lor \neg \left(z \leq 5 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \frac{1 + \left(y - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(y - z\right)\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-46} \lor \neg \left(z \leq 1.7 \cdot 10^{-75}\right):\\ \;\;\;\;x\_m \cdot \frac{1 + \left(y - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + y}{\frac{z}{x\_m}}\\ \end{array} \end{array} \]

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 (<= z -6.2e-46) (not (<= z 1.7e-75)))
    (* x_m (/ (+ 1.0 (- y z)) z))
    (/ (+ 1.0 y) (/ z x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -6.2e-46) || !(z <= 1.7e-75)) {
		tmp = x_m * ((1.0 + (y - z)) / z);
	} else {
		tmp = (1.0 + y) / (z / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-6.2d-46)) .or. (.not. (z <= 1.7d-75))) then
        tmp = x_m * ((1.0d0 + (y - z)) / z)
    else
        tmp = (1.0d0 + y) / (z / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -6.2e-46) || !(z <= 1.7e-75)) {
		tmp = x_m * ((1.0 + (y - z)) / z);
	} else {
		tmp = (1.0 + y) / (z / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (z <= -6.2e-46) or not (z <= 1.7e-75):
		tmp = x_m * ((1.0 + (y - z)) / z)
	else:
		tmp = (1.0 + y) / (z / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((z <= -6.2e-46) || !(z <= 1.7e-75))
		tmp = Float64(x_m * Float64(Float64(1.0 + Float64(y - z)) / z));
	else
		tmp = Float64(Float64(1.0 + y) / Float64(z / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z <= -6.2e-46) || ~((z <= 1.7e-75)))
		tmp = x_m * ((1.0 + (y - z)) / z);
	else
		tmp = (1.0 + 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[Or[LessEqual[z, -6.2e-46], N[Not[LessEqual[z, 1.7e-75]], $MachinePrecision]], N[(x$95$m * N[(N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + y), $MachinePrecision] / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -6.2000000000000002e-46 or 1.70000000000000008e-75 < z

   1. Initial program 75.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   if -6.2000000000000002e-46 < z < 1.70000000000000008e-75

   1. Initial program 99.9%

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

      1. associate-/l* 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*l/ 99.9%

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

      4. +-commutative 99.9%

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

   7. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. clear-num 99.6%

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

      4. un-div-inv 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. +-commutative 99.8%

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

      8. associate--l+ 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in z around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-46} \lor \neg \left(z \leq 1.7 \cdot 10^{-75}\right):\\ \;\;\;\;x \cdot \frac{1 + \left(y - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + y}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+75} \lor \neg \left(z \leq 2.1 \cdot 10^{+102}\right):\\ \;\;\;\;-x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + y}{\frac{z}{x\_m}}\\ \end{array} \end{array} \]

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 (<= z -3.9e+75) (not (<= z 2.1e+102)))
    (- x_m)
    (/ (+ 1.0 y) (/ z x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -3.9e+75) || !(z <= 2.1e+102)) {
		tmp = -x_m;
	} else {
		tmp = (1.0 + y) / (z / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-3.9d+75)) .or. (.not. (z <= 2.1d+102))) then
        tmp = -x_m
    else
        tmp = (1.0d0 + y) / (z / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -3.9e+75) || !(z <= 2.1e+102)) {
		tmp = -x_m;
	} else {
		tmp = (1.0 + y) / (z / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (z <= -3.9e+75) or not (z <= 2.1e+102):
		tmp = -x_m
	else:
		tmp = (1.0 + y) / (z / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((z <= -3.9e+75) || !(z <= 2.1e+102))
		tmp = Float64(-x_m);
	else
		tmp = Float64(Float64(1.0 + y) / Float64(z / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z <= -3.9e+75) || ~((z <= 2.1e+102)))
		tmp = -x_m;
	else
		tmp = (1.0 + 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[Or[LessEqual[z, -3.9e+75], N[Not[LessEqual[z, 2.1e+102]], $MachinePrecision]], (-x$95$m), N[(N[(1.0 + y), $MachinePrecision] / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.90000000000000038e75 or 2.10000000000000001e102 < z

   1. Initial program 64.8%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 81.4%

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

      1. neg-mul-1 81.4%

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

   7. Simplified 81.4%

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

   if -3.90000000000000038e75 < z < 2.10000000000000001e102

   1. Initial program 98.1%

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

      1. associate-/l* 93.7%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in x around 0 98.1%

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

      1. associate--l+ 98.1%

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

      2. +-commutative 98.1%

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

      3. associate-*l/ 99.1%

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

      4. +-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. +-commutative 99.1%

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

      2. *-commutative 99.1%

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

      3. clear-num 98.4%

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

      4. un-div-inv 98.5%

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

      5. +-commutative 98.5%

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

      6. associate-+r- 98.5%

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

      7. +-commutative 98.5%

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

      8. associate--l+ 98.5%

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

   9. Applied egg-rr 98.5%

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

   10. Taylor expanded in z around 0 93.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+75} \lor \neg \left(z \leq 2.1 \cdot 10^{+102}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + y}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+77} \lor \neg \left(z \leq 1.7 \cdot 10^{+102}\right):\\ \;\;\;\;-x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

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 (or (<= z -1.7e+77) (not (<= z 1.7e+102))) (- 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 ((z <= -1.7e+77) || !(z <= 1.7e+102)) {
		tmp = -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 ((z <= (-1.7d+77)) .or. (.not. (z <= 1.7d+102))) then
        tmp = -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 ((z <= -1.7e+77) || !(z <= 1.7e+102)) {
		tmp = -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 (z <= -1.7e+77) or not (z <= 1.7e+102):
		tmp = -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 ((z <= -1.7e+77) || !(z <= 1.7e+102))
		tmp = Float64(-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 ((z <= -1.7e+77) || ~((z <= 1.7e+102)))
		tmp = -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[Or[LessEqual[z, -1.7e+77], N[Not[LessEqual[z, 1.7e+102]], $MachinePrecision]], (-x$95$m), N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.69999999999999998e77 or 1.7e102 < z

   1. Initial program 64.8%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 81.4%

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

      1. neg-mul-1 81.4%

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

   7. Simplified 81.4%

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

   if -1.69999999999999998e77 < z < 1.7e102

   1. Initial program 98.1%

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

      1. associate-/l* 93.7%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in y around inf 53.6%

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

      1. *-commutative 53.6%

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

      2. associate-/l* 58.8%

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

   7. Applied egg-rr 58.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+77} \lor \neg \left(z \leq 1.7 \cdot 10^{+102}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -112000000:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+84}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x\_m}}\\ \end{array} \end{array} \]

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 -112000000.0)
    (/ (* x_m y) z)
    (if (<= y 2.7e+84) (- (/ x_m z) x_m) (/ y (/ z x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -112000000.0) {
		tmp = (x_m * y) / z;
	} else if (y <= 2.7e+84) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = 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 <= (-112000000.0d0)) then
        tmp = (x_m * y) / z
    else if (y <= 2.7d+84) then
        tmp = (x_m / z) - x_m
    else
        tmp = 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 <= -112000000.0) {
		tmp = (x_m * y) / z;
	} else if (y <= 2.7e+84) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = 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 <= -112000000.0:
		tmp = (x_m * y) / z
	elif y <= 2.7e+84:
		tmp = (x_m / z) - x_m
	else:
		tmp = 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 <= -112000000.0)
		tmp = Float64(Float64(x_m * y) / z);
	elseif (y <= 2.7e+84)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(y / Float64(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 <= -112000000.0)
		tmp = (x_m * y) / z;
	elseif (y <= 2.7e+84)
		tmp = (x_m / z) - x_m;
	else
		tmp = 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, -112000000.0], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 2.7e+84], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(y / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.12e8

   1. Initial program 90.0%

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

      1. associate-/l* 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around inf 73.2%

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

   if -1.12e8 < y < 2.7e84

   1. Initial program 81.4%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 76.3%

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

   6. Taylor expanded in z around inf 94.8%

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

      1. neg-mul-1 94.8%

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

      2. +-commutative 94.8%

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

      3. unsub-neg 94.8%

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

   8. Simplified 94.8%

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

   if 2.7e84 < y

   1. Initial program 89.1%

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

      1. associate-/l* 91.0%

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

   3. Simplified 91.0%

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

      1. clear-num 90.9%

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

      2. un-div-inv 92.5%

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

   6. Applied egg-rr 92.5%

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

      1. clear-num 90.9%

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

      2. inv-pow 90.9%

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

      3. +-commutative 90.9%

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

      4. associate-+r- 90.9%

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

      5. +-commutative 90.9%

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

      6. associate--l+ 90.9%

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

   8. Applied egg-rr 90.9%

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

      1. unpow-1 90.9%

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

      2. +-commutative 90.9%

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

      3. associate-+l- 90.9%

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

   10. Simplified 90.9%

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

   11. Taylor expanded in y around inf 80.3%

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

       1. associate-*r/ 77.7%

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

       2. *-commutative 77.7%

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

       3. associate-/r/ 84.2%

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

   13. Simplified 84.2%

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

Alternative 8: 84.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -57000000:\\ \;\;\;\;\frac{x\_m}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x\_m}}\\ \end{array} \end{array} \]

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 -57000000.0)
    (/ x_m (/ z y))
    (if (<= y 5.5e+83) (- (/ x_m z) x_m) (/ y (/ z x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -57000000.0) {
		tmp = x_m / (z / y);
	} else if (y <= 5.5e+83) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = 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 <= (-57000000.0d0)) then
        tmp = x_m / (z / y)
    else if (y <= 5.5d+83) then
        tmp = (x_m / z) - x_m
    else
        tmp = 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 <= -57000000.0) {
		tmp = x_m / (z / y);
	} else if (y <= 5.5e+83) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = 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 <= -57000000.0:
		tmp = x_m / (z / y)
	elif y <= 5.5e+83:
		tmp = (x_m / z) - x_m
	else:
		tmp = 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 <= -57000000.0)
		tmp = Float64(x_m / Float64(z / y));
	elseif (y <= 5.5e+83)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(y / Float64(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 <= -57000000.0)
		tmp = x_m / (z / y);
	elseif (y <= 5.5e+83)
		tmp = (x_m / z) - x_m;
	else
		tmp = 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, -57000000.0], N[(x$95$m / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e+83], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(y / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -5.7e7

   1. Initial program 90.0%

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

      1. associate-/l* 92.4%

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

   3. Simplified 92.4%

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

      1. clear-num 92.3%

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

      2. un-div-inv 93.6%

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

   6. Applied egg-rr 93.6%

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

   7. Taylor expanded in y around inf 69.5%

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

   if -5.7e7 < y < 5.4999999999999996e83

   1. Initial program 81.4%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 76.3%

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

   6. Taylor expanded in z around inf 94.8%

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

      1. neg-mul-1 94.8%

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

      2. +-commutative 94.8%

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

      3. unsub-neg 94.8%

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

   8. Simplified 94.8%

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

   if 5.4999999999999996e83 < y

   1. Initial program 89.1%

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

      1. associate-/l* 91.0%

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

   3. Simplified 91.0%

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

      1. clear-num 90.9%

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

      2. un-div-inv 92.5%

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

   6. Applied egg-rr 92.5%

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

      1. clear-num 90.9%

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

      2. inv-pow 90.9%

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

      3. +-commutative 90.9%

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

      4. associate-+r- 90.9%

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

      5. +-commutative 90.9%

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

      6. associate--l+ 90.9%

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

   8. Applied egg-rr 90.9%

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

      1. unpow-1 90.9%

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

      2. +-commutative 90.9%

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

      3. associate-+l- 90.9%

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

   10. Simplified 90.9%

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

   11. Taylor expanded in y around inf 80.3%

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

       1. associate-*r/ 77.7%

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

       2. *-commutative 77.7%

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

       3. associate-/r/ 84.2%

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

   13. Simplified 84.2%

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

Alternative 9: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -90000000:\\ \;\;\;\;\frac{x\_m}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+84}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

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 -90000000.0)
    (/ x_m (/ z y))
    (if (<= y 1.35e+84) (- (/ 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 <= -90000000.0) {
		tmp = x_m / (z / y);
	} else if (y <= 1.35e+84) {
		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 <= (-90000000.0d0)) then
        tmp = x_m / (z / y)
    else if (y <= 1.35d+84) 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 <= -90000000.0) {
		tmp = x_m / (z / y);
	} else if (y <= 1.35e+84) {
		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 <= -90000000.0:
		tmp = x_m / (z / y)
	elif y <= 1.35e+84:
		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 <= -90000000.0)
		tmp = Float64(x_m / Float64(z / y));
	elseif (y <= 1.35e+84)
		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 <= -90000000.0)
		tmp = x_m / (z / y);
	elseif (y <= 1.35e+84)
		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, -90000000.0], N[(x$95$m / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+84], 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 < -9e7

   1. Initial program 90.0%

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

      1. associate-/l* 92.4%

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

   3. Simplified 92.4%

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

      1. clear-num 92.3%

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

      2. un-div-inv 93.6%

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

   6. Applied egg-rr 93.6%

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

   7. Taylor expanded in y around inf 69.5%

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

   if -9e7 < y < 1.35e84

   1. Initial program 81.4%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 76.3%

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

   6. Taylor expanded in z around inf 94.8%

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

      1. neg-mul-1 94.8%

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

      2. +-commutative 94.8%

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

      3. unsub-neg 94.8%

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

   8. Simplified 94.8%

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

   if 1.35e84 < y

   1. Initial program 89.1%

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

      1. associate-/l* 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in y around inf 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-/l* 84.2%

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

   7. Applied egg-rr 84.2%

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

Alternative 10: 84.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -50000000:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+90}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

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 -50000000.0)
    (* x_m (/ y z))
    (if (<= y 1.25e+90) (- (/ 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 <= -50000000.0) {
		tmp = x_m * (y / z);
	} else if (y <= 1.25e+90) {
		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 <= (-50000000.0d0)) then
        tmp = x_m * (y / z)
    else if (y <= 1.25d+90) 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 <= -50000000.0) {
		tmp = x_m * (y / z);
	} else if (y <= 1.25e+90) {
		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 <= -50000000.0:
		tmp = x_m * (y / z)
	elif y <= 1.25e+90:
		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 <= -50000000.0)
		tmp = Float64(x_m * Float64(y / z));
	elseif (y <= 1.25e+90)
		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 <= -50000000.0)
		tmp = x_m * (y / z);
	elseif (y <= 1.25e+90)
		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, -50000000.0], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+90], 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 < -5e7

   1. Initial program 90.0%

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

      1. associate-/l* 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around inf 73.2%

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

      1. associate-/l* 68.3%

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

   7. Simplified 68.3%

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

   if -5e7 < y < 1.2500000000000001e90

   1. Initial program 81.4%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 76.3%

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

   6. Taylor expanded in z around inf 94.8%

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

      1. neg-mul-1 94.8%

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

      2. +-commutative 94.8%

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

      3. unsub-neg 94.8%

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

   8. Simplified 94.8%

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

   if 1.2500000000000001e90 < y

   1. Initial program 89.1%

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

      1. associate-/l* 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in y around inf 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-/l* 84.2%

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

   7. Applied egg-rr 84.2%

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

Alternative 11: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := 1 + \left(y - z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3 \cdot 10^{-62}:\\ \;\;\;\;\frac{x\_m \cdot t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{t\_0}}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (- y z))))
   (* x_s (if (<= x_m 3e-62) (/ (* x_m t_0) z) (/ x_m (/ z t_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 t_0 = 1.0 + (y - z);
	double tmp;
	if (x_m <= 3e-62) {
		tmp = (x_m * t_0) / z;
	} else {
		tmp = x_m / (z / t_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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (y - z)
    if (x_m <= 3d-62) then
        tmp = (x_m * t_0) / z
    else
        tmp = x_m / (z / t_0)
    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 t_0 = 1.0 + (y - z);
	double tmp;
	if (x_m <= 3e-62) {
		tmp = (x_m * t_0) / z;
	} else {
		tmp = x_m / (z / t_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):
	t_0 = 1.0 + (y - z)
	tmp = 0
	if x_m <= 3e-62:
		tmp = (x_m * t_0) / z
	else:
		tmp = x_m / (z / t_0)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(1.0 + Float64(y - z))
	tmp = 0.0
	if (x_m <= 3e-62)
		tmp = Float64(Float64(x_m * t_0) / z);
	else
		tmp = Float64(x_m / Float64(z / t_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)
	t_0 = 1.0 + (y - z);
	tmp = 0.0;
	if (x_m <= 3e-62)
		tmp = (x_m * t_0) / z;
	else
		tmp = x_m / (z / t_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_] := Block[{t$95$0 = N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 3e-62], N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.0000000000000001e-62

   1. Initial program 88.6%

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

   if 3.0000000000000001e-62 < x

   1. Initial program 78.1%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-62}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.7% 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 71.0%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 68.9%

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

      1. neg-mul-1 68.9%

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

   7. Simplified 68.9%

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

   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* 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in z around 0 98.8%

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

      1. associate-/l* 91.2%

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

   7. Simplified 91.2%

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

   8. Taylor expanded in y around 0 55.2%

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

4. Final simplification 62.2%

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

Alternative 13: 96.3% accurate, 1.0× speedup

Math

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

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 (/ z (+ 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) {
	return x_s * (x_m / (z / (1.0 + (y - z))));
}

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 / (z / (1.0d0 + (y - z))))
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 / (z / (1.0 + (y - z))));
}

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

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(z / Float64(1.0 + Float64(y - z)))))
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 / (z / (1.0 + (y - z))));
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[(z / N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.2%

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

   1. associate-/l* 96.1%

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

3. Simplified 96.1%

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

   1. clear-num 96.1%

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

   2. un-div-inv 96.8%

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

6. Applied egg-rr 96.8%

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

7. Final simplification 96.8%

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

Alternative 14: 39.4% 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 85.2%

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

   1. associate-/l* 96.1%

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

3. Simplified 96.1%

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

5. Taylor expanded in z around inf 36.6%

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

   1. neg-mul-1 36.6%

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

7. Simplified 36.6%

   \[\leadsto \color{blue}{-x} \]
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 85.2%

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

   1. associate-/l* 96.1%

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

3. Simplified 96.1%

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

5. Taylor expanded in z around inf 36.6%

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

   1. neg-mul-1 36.6%

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

7. Simplified 36.6%

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

   1. neg-sub0 36.6%

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

   2. sub-neg 36.6%

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

   3. add-sqr-sqrt 16.6%

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

   4. sqrt-unprod 14.1%

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

   5. sqr-neg 14.1%

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

   6. sqrt-unprod 1.4%

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

   7. add-sqr-sqrt 3.0%

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

9. Applied egg-rr 3.0%

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

    1. +-lft-identity 3.0%

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

11. Simplified 3.0%

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

Developer Target 1: 99.1% 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 2024150 
(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))