Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.1% → 97.8%

Time: 7.3s

Alternatives: 8

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * (y + z)) / 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)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * (y + z)) / 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)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \left(y + z\right)}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{+302}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x\_m + x\_m \cdot \frac{y}{z}\\ \end{array} \end{array} \end{array} \]

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 (/ (* x_m (+ y z)) z)))
   (*
    x_s
    (if (<= t_0 0.0)
      (* (+ y z) (/ x_m z))
      (if (<= t_0 1e+302) t_0 (+ x_m (* x_m (/ 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 t_0 = (x_m * (y + z)) / z;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (y + z) * (x_m / z);
	} else if (t_0 <= 1e+302) {
		tmp = t_0;
	} else {
		tmp = x_m + (x_m * (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) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (y + z)) / z
    if (t_0 <= 0.0d0) then
        tmp = (y + z) * (x_m / z)
    else if (t_0 <= 1d+302) then
        tmp = t_0
    else
        tmp = x_m + (x_m * (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 t_0 = (x_m * (y + z)) / z;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (y + z) * (x_m / z);
	} else if (t_0 <= 1e+302) {
		tmp = t_0;
	} else {
		tmp = x_m + (x_m * (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):
	t_0 = (x_m * (y + z)) / z
	tmp = 0
	if t_0 <= 0.0:
		tmp = (y + z) * (x_m / z)
	elif t_0 <= 1e+302:
		tmp = t_0
	else:
		tmp = x_m + (x_m * (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)
	t_0 = Float64(Float64(x_m * Float64(y + z)) / z)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(y + z) * Float64(x_m / z));
	elseif (t_0 <= 1e+302)
		tmp = t_0;
	else
		tmp = Float64(x_m + Float64(x_m * 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)
	t_0 = (x_m * (y + z)) / z;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (y + z) * (x_m / z);
	elseif (t_0 <= 1e+302)
		tmp = t_0;
	else
		tmp = x_m + (x_m * (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_] := Block[{t$95$0 = N[(N[(x$95$m * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(y + z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+302], t$95$0, N[(x$95$m + N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < 0.0

   1. Initial program 78.6%

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

      1. *-commutative 78.6%

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

      2. associate-/l* 89.9%

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

   3. Simplified 89.9%

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

   if 0.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < 1.0000000000000001e302

   1. Initial program 99.7%

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

   if 1.0000000000000001e302 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 59.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. div-sub 100.0%

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

      5. remove-double-neg 100.0%

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

      6. distribute-frac-neg2 100.0%

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

      7. *-inverses 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-in 99.9%

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

      4. *-un-lft-identity 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq 0:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 10^{+302}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-145} \lor \neg \left(z \leq 1.65 \cdot 10^{-165}\right):\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= z -2.9e-145) (not (<= z 1.65e-165)))
    (* x_m (- (/ y z) -1.0))
    (* (+ y z) (/ x_m z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -2.9e-145) || !(z <= 1.65e-165)) {
		tmp = x_m * ((y / z) - -1.0);
	} else {
		tmp = (y + z) * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.9d-145)) .or. (.not. (z <= 1.65d-165))) then
        tmp = x_m * ((y / z) - (-1.0d0))
    else
        tmp = (y + z) * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -2.9e-145) || !(z <= 1.65e-165)) {
		tmp = x_m * ((y / z) - -1.0);
	} else {
		tmp = (y + z) * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (z <= -2.9e-145) or not (z <= 1.65e-165):
		tmp = x_m * ((y / z) - -1.0)
	else:
		tmp = (y + z) * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((z <= -2.9e-145) || !(z <= 1.65e-165))
		tmp = Float64(x_m * Float64(Float64(y / z) - -1.0));
	else
		tmp = Float64(Float64(y + z) * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z <= -2.9e-145) || ~((z <= 1.65e-165)))
		tmp = x_m * ((y / z) - -1.0);
	else
		tmp = (y + z) * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[z, -2.9e-145], N[Not[LessEqual[z, 1.65e-165]], $MachinePrecision]], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y + z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.89999999999999984e-145 or 1.6499999999999999e-165 < z

   1. Initial program 82.3%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. remove-double-neg 99.9%

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

      6. distribute-frac-neg2 99.9%

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

      7. *-inverses 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   if -2.89999999999999984e-145 < z < 1.6499999999999999e-165

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-/l* 95.3%

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

   3. Simplified 95.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-145} \lor \neg \left(z \leq 1.65 \cdot 10^{-165}\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-122} \lor \neg \left(z \leq 7.5 \cdot 10^{-185}\right):\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x\_m}}\\ \end{array} \end{array} \]

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 (or (<= z -6.5e-122) (not (<= z 7.5e-185)))
    (* x_m (- (/ y 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 <= -6.5e-122) || !(z <= 7.5e-185)) {
		tmp = x_m * ((y / z) - -1.0);
	} 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 ((z <= (-6.5d-122)) .or. (.not. (z <= 7.5d-185))) then
        tmp = x_m * ((y / z) - (-1.0d0))
    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 ((z <= -6.5e-122) || !(z <= 7.5e-185)) {
		tmp = x_m * ((y / z) - -1.0);
	} 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 (z <= -6.5e-122) or not (z <= 7.5e-185):
		tmp = x_m * ((y / z) - -1.0)
	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 ((z <= -6.5e-122) || !(z <= 7.5e-185))
		tmp = Float64(x_m * Float64(Float64(y / z) - -1.0));
	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 ((z <= -6.5e-122) || ~((z <= 7.5e-185)))
		tmp = x_m * ((y / z) - -1.0);
	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[Or[LessEqual[z, -6.5e-122], N[Not[LessEqual[z, 7.5e-185]], $MachinePrecision]], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -6.49999999999999965e-122 or 7.49999999999999978e-185 < z

   1. Initial program 82.3%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. remove-double-neg 99.9%

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

      6. distribute-frac-neg2 99.9%

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

      7. *-inverses 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   if -6.49999999999999965e-122 < z < 7.49999999999999978e-185

   1. Initial program 89.4%

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

   3. Taylor expanded in y around inf 85.1%

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

      1. *-commutative 85.1%

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

   5. Simplified 85.1%

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

      1. associate-*l/ 69.3%

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

   7. Applied egg-rr 69.3%

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

      1. associate-*l/ 85.1%

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

      2. associate-*r/ 90.9%

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

      3. clear-num 90.8%

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

      4. un-div-inv 91.1%

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

   9. Applied egg-rr 91.1%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-122} \lor \neg \left(z \leq 7.5 \cdot 10^{-185}\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-145}:\\ \;\;\;\;x\_m + x\_m \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-166}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - -1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -1.8e-145)
    (+ x_m (* x_m (/ y z)))
    (if (<= z 5.2e-166) (* (+ y z) (/ x_m z)) (* x_m (- (/ y z) -1.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1.8e-145) {
		tmp = x_m + (x_m * (y / z));
	} else if (z <= 5.2e-166) {
		tmp = (y + z) * (x_m / z);
	} else {
		tmp = x_m * ((y / z) - -1.0);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.8d-145)) then
        tmp = x_m + (x_m * (y / z))
    else if (z <= 5.2d-166) then
        tmp = (y + z) * (x_m / z)
    else
        tmp = x_m * ((y / z) - (-1.0d0))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1.8e-145) {
		tmp = x_m + (x_m * (y / z));
	} else if (z <= 5.2e-166) {
		tmp = (y + z) * (x_m / z);
	} else {
		tmp = x_m * ((y / z) - -1.0);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= -1.8e-145:
		tmp = x_m + (x_m * (y / z))
	elif z <= 5.2e-166:
		tmp = (y + z) * (x_m / z)
	else:
		tmp = x_m * ((y / z) - -1.0)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -1.8e-145)
		tmp = Float64(x_m + Float64(x_m * Float64(y / z)));
	elseif (z <= 5.2e-166)
		tmp = Float64(Float64(y + z) * Float64(x_m / z));
	else
		tmp = Float64(x_m * Float64(Float64(y / z) - -1.0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -1.8e-145)
		tmp = x_m + (x_m * (y / z));
	elseif (z <= 5.2e-166)
		tmp = (y + z) * (x_m / z);
	else
		tmp = x_m * ((y / z) - -1.0);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -1.8e-145], N[(x$95$m + N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.2e-166], N[(N[(y + z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.8e-145

   1. Initial program 85.7%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. remove-double-neg 99.9%

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

      6. distribute-frac-neg2 99.9%

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

      7. *-inverses 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. *-un-lft-identity 99.9%

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

   6. Applied egg-rr 99.9%

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

   if -1.8e-145 < z < 5.19999999999999979e-166

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-/l* 95.3%

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

   3. Simplified 95.3%

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

   if 5.19999999999999979e-166 < z

   1. Initial program 79.5%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. remove-double-neg 99.9%

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

      6. distribute-frac-neg2 99.9%

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

      7. *-inverses 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-145}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-166}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+15}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{+59}:\\ \;\;\;\;\frac{y}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -6e+15) x_m (if (<= z 3.05e+59) (/ y (/ z x_m)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -6e+15) {
		tmp = x_m;
	} else if (z <= 3.05e+59) {
		tmp = y / (z / x_m);
	} 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 <= (-6d+15)) then
        tmp = x_m
    else if (z <= 3.05d+59) then
        tmp = y / (z / x_m)
    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 <= -6e+15) {
		tmp = x_m;
	} else if (z <= 3.05e+59) {
		tmp = y / (z / x_m);
	} 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 <= -6e+15:
		tmp = x_m
	elif z <= 3.05e+59:
		tmp = y / (z / x_m)
	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 <= -6e+15)
		tmp = x_m;
	elseif (z <= 3.05e+59)
		tmp = Float64(y / Float64(z / x_m));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -6e+15)
		tmp = x_m;
	elseif (z <= 3.05e+59)
		tmp = y / (z / x_m);
	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, -6e+15], x$95$m, If[LessEqual[z, 3.05e+59], N[(y / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -6e15 or 3.04999999999999986e59 < z

   1. Initial program 71.5%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. remove-double-neg 99.9%

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

      6. distribute-frac-neg2 99.9%

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

      7. *-inverses 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 82.5%

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

   if -6e15 < z < 3.04999999999999986e59

   1. Initial program 92.9%

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

   3. Taylor expanded in y around inf 75.0%

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

      1. *-commutative 75.0%

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

   5. Simplified 75.0%

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

      1. associate-*l/ 69.2%

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

   7. Applied egg-rr 69.2%

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

      1. associate-*l/ 75.0%

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

      2. associate-*r/ 78.5%

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

      3. clear-num 78.4%

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

      4. un-div-inv 78.6%

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

   9. Applied egg-rr 78.6%

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

Alternative 6: 72.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+14}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+58}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -6.2e+14) x_m (if (<= z 4.5e+58) (* y (/ 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 <= -6.2e+14) {
		tmp = x_m;
	} else if (z <= 4.5e+58) {
		tmp = y * (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 <= (-6.2d+14)) then
        tmp = x_m
    else if (z <= 4.5d+58) then
        tmp = y * (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 <= -6.2e+14) {
		tmp = x_m;
	} else if (z <= 4.5e+58) {
		tmp = y * (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 <= -6.2e+14:
		tmp = x_m
	elif z <= 4.5e+58:
		tmp = y * (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 <= -6.2e+14)
		tmp = x_m;
	elseif (z <= 4.5e+58)
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = 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+14)
		tmp = x_m;
	elseif (z <= 4.5e+58)
		tmp = y * (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, -6.2e+14], x$95$m, If[LessEqual[z, 4.5e+58], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -6.2e14 or 4.4999999999999998e58 < z

   1. Initial program 71.5%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. remove-double-neg 99.9%

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

      6. distribute-frac-neg2 99.9%

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

      7. *-inverses 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 82.5%

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

   if -6.2e14 < z < 4.4999999999999998e58

   1. Initial program 92.9%

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

      1. associate-/l* 89.8%

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

      2. remove-double-neg 89.8%

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

      3. unsub-neg 89.8%

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

      4. div-sub 89.9%

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

      5. remove-double-neg 89.9%

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

      6. distribute-frac-neg2 89.9%

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

      7. *-inverses 89.9%

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

      8. metadata-eval 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in y around inf 75.0%

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

      1. associate-*l/ 78.5%

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

      2. *-commutative 78.5%

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

   7. Simplified 78.5%

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

Alternative 7: 70.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+15}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+59}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -5.4e+15) x_m (if (<= z 4.2e+59) (* x_m (/ y z)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -5.4e+15) {
		tmp = x_m;
	} else if (z <= 4.2e+59) {
		tmp = x_m * (y / z);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-5.4d+15)) then
        tmp = x_m
    else if (z <= 4.2d+59) then
        tmp = x_m * (y / z)
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -5.4e+15) {
		tmp = x_m;
	} else if (z <= 4.2e+59) {
		tmp = x_m * (y / z);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= -5.4e+15:
		tmp = x_m
	elif z <= 4.2e+59:
		tmp = x_m * (y / z)
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -5.4e+15)
		tmp = x_m;
	elseif (z <= 4.2e+59)
		tmp = Float64(x_m * Float64(y / z));
	else
		tmp = 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 <= -5.4e+15)
		tmp = x_m;
	elseif (z <= 4.2e+59)
		tmp = x_m * (y / z);
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -5.4e+15], x$95$m, If[LessEqual[z, 4.2e+59], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -5.4e15 or 4.19999999999999968e59 < z

   1. Initial program 71.5%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. remove-double-neg 99.9%

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

      6. distribute-frac-neg2 99.9%

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

      7. *-inverses 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 82.5%

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

   if -5.4e15 < z < 4.19999999999999968e59

   1. Initial program 92.9%

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

      1. associate-/l* 89.8%

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

      2. remove-double-neg 89.8%

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

      3. unsub-neg 89.8%

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

      4. div-sub 89.9%

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

      5. remove-double-neg 89.9%

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

      6. distribute-frac-neg2 89.9%

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

      7. *-inverses 89.9%

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

      8. metadata-eval 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in y around inf 69.2%

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

Alternative 8: 51.2% accurate, 7.0× 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 84.1%

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

   1. associate-/l* 94.0%

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

   2. remove-double-neg 94.0%

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

   3. unsub-neg 94.0%

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

   4. div-sub 94.0%

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

   5. remove-double-neg 94.0%

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

   6. distribute-frac-neg2 94.0%

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

   7. *-inverses 94.0%

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

   8. metadata-eval 94.0%

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

3. Simplified 94.0%

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

5. Taylor expanded in y around 0 47.2%

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

Developer Target 1: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{z}{y + z}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ x (/ z (+ y z))))

C

double code(double x, double y, double z) {
	return x / (z / (y + 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 / (z / (y + z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024165 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (/ x (/ z (+ y z))))

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