Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.8% → 96.3%

Time: 5.3s

Alternatives: 8

Speedup: 1.0×

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: 84.8% 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: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.02 \cdot 10^{+21}:\\ \;\;\;\;\frac{x\_m \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m + x\_m \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= x_m 1.02e+21) (/ (* x_m (+ y z)) z) (+ 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 tmp;
	if (x_m <= 1.02e+21) {
		tmp = (x_m * (y + z)) / z;
	} 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) :: tmp
    if (x_m <= 1.02d+21) then
        tmp = (x_m * (y + z)) / z
    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 tmp;
	if (x_m <= 1.02e+21) {
		tmp = (x_m * (y + z)) / z;
	} 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):
	tmp = 0
	if x_m <= 1.02e+21:
		tmp = (x_m * (y + z)) / z
	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)
	tmp = 0.0
	if (x_m <= 1.02e+21)
		tmp = Float64(Float64(x_m * Float64(y + z)) / z);
	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)
	tmp = 0.0;
	if (x_m <= 1.02e+21)
		tmp = (x_m * (y + z)) / z;
	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_] := N[(x$95$s * If[LessEqual[x$95$m, 1.02e+21], N[(N[(x$95$m * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m + N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.02e21

   1. Initial program 93.6%

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

   if 1.02e21 < x

   1. Initial program 67.1%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 100.0%

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

      4. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+21}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-107}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-277}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+29}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -7e-107)
    x_m
    (if (<= z -5.2e-277)
      (* y (/ x_m z))
      (if (<= z 3.1e+29) (* 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 <= -7e-107) {
		tmp = x_m;
	} else if (z <= -5.2e-277) {
		tmp = y * (x_m / z);
	} else if (z <= 3.1e+29) {
		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 <= (-7d-107)) then
        tmp = x_m
    else if (z <= (-5.2d-277)) then
        tmp = y * (x_m / z)
    else if (z <= 3.1d+29) 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 <= -7e-107) {
		tmp = x_m;
	} else if (z <= -5.2e-277) {
		tmp = y * (x_m / z);
	} else if (z <= 3.1e+29) {
		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 <= -7e-107:
		tmp = x_m
	elif z <= -5.2e-277:
		tmp = y * (x_m / z)
	elif z <= 3.1e+29:
		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 <= -7e-107)
		tmp = x_m;
	elseif (z <= -5.2e-277)
		tmp = Float64(y * Float64(x_m / z));
	elseif (z <= 3.1e+29)
		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 <= -7e-107)
		tmp = x_m;
	elseif (z <= -5.2e-277)
		tmp = y * (x_m / z);
	elseif (z <= 3.1e+29)
		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, -7e-107], x$95$m, If[LessEqual[z, -5.2e-277], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.1e+29], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], x$95$m]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -6.99999999999999971e-107 or 3.0999999999999999e29 < z

   1. Initial program 77.0%

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

      1. associate-/l* 98.7%

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

      2. remove-double-neg 98.7%

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

      3. unsub-neg 98.7%

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

      4. div-sub 98.7%

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

      5. remove-double-neg 98.7%

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

      6. distribute-frac-neg2 98.7%

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

      7. *-inverses 98.7%

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

      8. metadata-eval 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in y around 0 76.3%

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

   if -6.99999999999999971e-107 < z < -5.2e-277

   1. Initial program 96.8%

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

      1. associate-/l* 84.5%

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

      2. remove-double-neg 84.5%

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

      3. unsub-neg 84.5%

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

      4. div-sub 84.5%

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

      5. remove-double-neg 84.5%

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

      6. distribute-frac-neg2 84.5%

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

      7. *-inverses 84.5%

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

      8. metadata-eval 84.5%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in y around inf 86.7%

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

      1. associate-*l/ 84.6%

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

      2. *-commutative 84.6%

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

   7. Simplified 84.6%

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

   if -5.2e-277 < z < 3.0999999999999999e29

   1. Initial program 98.5%

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

      1. associate-/l* 94.9%

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

      2. remove-double-neg 94.9%

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

      3. unsub-neg 94.9%

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

      4. div-sub 94.9%

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

      5. remove-double-neg 94.9%

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

      6. distribute-frac-neg2 94.9%

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

      7. *-inverses 94.9%

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

      8. metadata-eval 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in y around inf 75.7%

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

Alternative 3: 72.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-107}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+31}:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -3.9e-107) x_m (if (<= z 4.2e+31) (/ (* 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 <= -3.9e-107) {
		tmp = x_m;
	} else if (z <= 4.2e+31) {
		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 <= (-3.9d-107)) then
        tmp = x_m
    else if (z <= 4.2d+31) 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 <= -3.9e-107) {
		tmp = x_m;
	} else if (z <= 4.2e+31) {
		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 <= -3.9e-107:
		tmp = x_m
	elif z <= 4.2e+31:
		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 <= -3.9e-107)
		tmp = x_m;
	elseif (z <= 4.2e+31)
		tmp = Float64(Float64(x_m * 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 <= -3.9e-107)
		tmp = x_m;
	elseif (z <= 4.2e+31)
		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, -3.9e-107], x$95$m, If[LessEqual[z, 4.2e+31], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.9000000000000001e-107 or 4.19999999999999958e31 < z

   1. Initial program 77.0%

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

      1. associate-/l* 98.7%

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

      2. remove-double-neg 98.7%

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

      3. unsub-neg 98.7%

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

      4. div-sub 98.7%

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

      5. remove-double-neg 98.7%

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

      6. distribute-frac-neg2 98.7%

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

      7. *-inverses 98.7%

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

      8. metadata-eval 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in y around 0 76.3%

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

   if -3.9000000000000001e-107 < z < 4.19999999999999958e31

   1. Initial program 98.0%

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

   3. Taylor expanded in y around inf 81.5%

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

      1. *-commutative 81.5%

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

   5. Simplified 81.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+31}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-107}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+30}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -2.8e-107) x_m (if (<= z 1.55e+30) (* 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 <= -2.8e-107) {
		tmp = x_m;
	} else if (z <= 1.55e+30) {
		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 <= (-2.8d-107)) then
        tmp = x_m
    else if (z <= 1.55d+30) 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 <= -2.8e-107) {
		tmp = x_m;
	} else if (z <= 1.55e+30) {
		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 <= -2.8e-107:
		tmp = x_m
	elif z <= 1.55e+30:
		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 <= -2.8e-107)
		tmp = x_m;
	elseif (z <= 1.55e+30)
		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 <= -2.8e-107)
		tmp = x_m;
	elseif (z <= 1.55e+30)
		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, -2.8e-107], x$95$m, If[LessEqual[z, 1.55e+30], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.7999999999999999e-107 or 1.5499999999999999e30 < z

   1. Initial program 77.0%

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

      1. associate-/l* 98.7%

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

      2. remove-double-neg 98.7%

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

      3. unsub-neg 98.7%

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

      4. div-sub 98.7%

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

      5. remove-double-neg 98.7%

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

      6. distribute-frac-neg2 98.7%

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

      7. *-inverses 98.7%

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

      8. metadata-eval 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in y around 0 76.3%

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

   if -2.7999999999999999e-107 < z < 1.5499999999999999e30

   1. Initial program 98.0%

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

      1. associate-/l* 91.8%

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

      2. remove-double-neg 91.8%

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

      3. unsub-neg 91.8%

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

      4. div-sub 91.8%

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

      5. remove-double-neg 91.8%

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

      6. distribute-frac-neg2 91.8%

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

      7. *-inverses 91.8%

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

      8. metadata-eval 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in y around inf 74.5%

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

Alternative 5: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85 \cdot 10^{-12}:\\ \;\;\;\;x\_m + \frac{y}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m + x\_m \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.85e-12) (+ x_m (/ y (/ z x_m))) (+ 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 tmp;
	if (x_m <= 1.85e-12) {
		tmp = x_m + (y / (z / x_m));
	} 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) :: tmp
    if (x_m <= 1.85d-12) then
        tmp = x_m + (y / (z / x_m))
    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 tmp;
	if (x_m <= 1.85e-12) {
		tmp = x_m + (y / (z / x_m));
	} 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):
	tmp = 0
	if x_m <= 1.85e-12:
		tmp = x_m + (y / (z / x_m))
	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)
	tmp = 0.0
	if (x_m <= 1.85e-12)
		tmp = Float64(x_m + Float64(y / Float64(z / x_m)));
	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)
	tmp = 0.0;
	if (x_m <= 1.85e-12)
		tmp = x_m + (y / (z / x_m));
	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_] := N[(x$95$s * If[LessEqual[x$95$m, 1.85e-12], N[(x$95$m + N[(y / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m + N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.84999999999999999e-12

   1. Initial program 93.3%

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

      1. associate-/l* 93.9%

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

      2. remove-double-neg 93.9%

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

      3. unsub-neg 93.9%

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

      4. div-sub 93.9%

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

      5. remove-double-neg 93.9%

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

      6. distribute-frac-neg2 93.9%

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

      7. *-inverses 93.9%

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

      8. metadata-eval 93.9%

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

   3. Simplified 93.9%

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

      1. sub-neg 93.9%

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

      2. metadata-eval 93.9%

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

      3. distribute-rgt-in 93.9%

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

      4. *-un-lft-identity 93.9%

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

   6. Applied egg-rr 93.9%

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

      1. associate-*l/ 97.1%

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

      2. associate-*r/ 92.7%

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

      3. clear-num 92.3%

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

      4. un-div-inv 93.1%

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

   8. Applied egg-rr 93.1%

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

   if 1.84999999999999999e-12 < x

   1. Initial program 69.9%

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

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.0% accurate, 1.0× speedup

Math

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

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 (* 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) {
	return x_s * (x_m + (x_m * (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 + (x_m * (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 + (x_m * (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 + (x_m * (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(x_m * 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 + (x_m * (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[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.6%

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

   1. associate-/l* 95.9%

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

   2. remove-double-neg 95.9%

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

   3. unsub-neg 95.9%

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

   4. div-sub 95.9%

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

   5. remove-double-neg 95.9%

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

   6. distribute-frac-neg2 95.9%

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

   7. *-inverses 95.9%

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

   8. metadata-eval 95.9%

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

3. Simplified 95.9%

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

   1. sub-neg 95.9%

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

   2. metadata-eval 95.9%

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

   3. distribute-rgt-in 95.9%

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

   4. *-un-lft-identity 95.9%

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

6. Applied egg-rr 95.9%

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

7. Final simplification 95.9%

   \[\leadsto x + x \cdot \frac{y}{z} \]
8. Add Preprocessing

Alternative 7: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

   1. associate-/l* 95.9%

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

   2. remove-double-neg 95.9%

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

   3. unsub-neg 95.9%

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

   4. div-sub 95.9%

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

   5. remove-double-neg 95.9%

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

   6. distribute-frac-neg2 95.9%

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

   7. *-inverses 95.9%

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

   8. metadata-eval 95.9%

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

3. Simplified 95.9%

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

Alternative 8: 51.4% 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 85.6%

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

   1. associate-/l* 95.9%

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

   2. remove-double-neg 95.9%

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

   3. unsub-neg 95.9%

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

   4. div-sub 95.9%

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

   5. remove-double-neg 95.9%

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

   6. distribute-frac-neg2 95.9%

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

   7. *-inverses 95.9%

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

   8. metadata-eval 95.9%

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

3. Simplified 95.9%

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

5. Taylor expanded in y around 0 52.5%

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

Developer Target 1: 96.4% 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 2024170 
(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))