Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 85.1% → 96.6%

Time: 7.1s

Alternatives: 9

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: 85.1% 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: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.4 \cdot 10^{-191}:\\ \;\;\;\;x\_m + \frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m + \frac{x\_m}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.4e-191) (+ x_m (/ (* x_m y) z)) (+ x_m (/ x_m (/ z y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.4e-191) {
		tmp = x_m + ((x_m * y) / z);
	} else {
		tmp = x_m + (x_m / (z / y));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 2.4d-191) then
        tmp = x_m + ((x_m * y) / z)
    else
        tmp = x_m + (x_m / (z / y))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.4e-191) {
		tmp = x_m + ((x_m * y) / z);
	} else {
		tmp = x_m + (x_m / (z / y));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 2.4e-191:
		tmp = x_m + ((x_m * y) / z)
	else:
		tmp = x_m + (x_m / (z / y))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 2.4e-191)
		tmp = Float64(x_m + Float64(Float64(x_m * y) / z));
	else
		tmp = Float64(x_m + Float64(x_m / Float64(z / y)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 2.4e-191)
		tmp = x_m + ((x_m * y) / z);
	else
		tmp = x_m + (x_m / (z / y));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 2.4e-191], N[(x$95$m + N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m + N[(x$95$m / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.3999999999999999e-191

   1. Initial program 89.7%

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

      1. associate-/l* 93.5%

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

      2. remove-double-neg 93.5%

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

      3. distribute-frac-neg2 93.5%

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

      4. neg-sub0 93.5%

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

      5. remove-double-neg 93.5%

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

      6. unsub-neg 93.5%

         \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

      7. div-sub 93.5%

         \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

      8. *-inverses 93.5%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

      9. metadata-eval 93.5%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

      10. associate--r- 93.5%

          \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

      11. neg-sub0 93.5%

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

      12. distribute-frac-neg2 93.5%

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

      13. remove-double-neg 93.5%

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

      14. sub-neg 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in y around 0 94.4%

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

   if 2.3999999999999999e-191 < x

   1. Initial program 78.8%

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

      1. associate-/l* 99.0%

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

      2. remove-double-neg 99.0%

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

      3. distribute-frac-neg2 99.0%

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

      4. neg-sub0 99.0%

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

      5. remove-double-neg 99.0%

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

      6. unsub-neg 99.0%

         \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

      7. div-sub 99.1%

         \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

      8. *-inverses 99.1%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

      9. metadata-eval 99.1%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

      10. associate--r- 99.1%

          \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

      11. neg-sub0 99.1%

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

      12. distribute-frac-neg2 99.1%

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

      13. remove-double-neg 99.1%

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

      14. sub-neg 99.1%

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

   3. Simplified 99.1%

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

      1. sub-neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. distribute-rgt-in 99.1%

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

      4. *-un-lft-identity 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. *-commutative 99.1%

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

      2. clear-num 99.0%

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

      3. un-div-inv 99.1%

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

   8. Applied egg-rr 99.1%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-191}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+97}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -3.4e+97) x_m (if (<= z 7.2e+16) (/ (* 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 <= -3.4e+97) {
		tmp = x_m;
	} else if (z <= 7.2e+16) {
		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 <= (-3.4d+97)) then
        tmp = x_m
    else if (z <= 7.2d+16) 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 <= -3.4e+97) {
		tmp = x_m;
	} else if (z <= 7.2e+16) {
		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 <= -3.4e+97:
		tmp = x_m
	elif z <= 7.2e+16:
		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 <= -3.4e+97)
		tmp = x_m;
	elseif (z <= 7.2e+16)
		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 <= -3.4e+97)
		tmp = x_m;
	elseif (z <= 7.2e+16)
		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, -3.4e+97], x$95$m, If[LessEqual[z, 7.2e+16], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.4000000000000001e97 or 7.2e16 < z

   1. Initial program 72.4%

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

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

      3. distribute-frac-neg2 100.0%

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

      4. neg-sub0 100.0%

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

      5. remove-double-neg 100.0%

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

      6. unsub-neg 100.0%

         \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

      7. div-sub 100.0%

         \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

      8. *-inverses 100.0%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

      9. metadata-eval 100.0%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

      10. associate--r- 100.0%

          \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. distribute-frac-neg2 100.0%

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

      13. remove-double-neg 100.0%

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

      14. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 81.2%

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

   if -3.4000000000000001e97 < z < 7.2e16

   1. Initial program 94.1%

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

   3. Taylor expanded in y around inf 72.6%

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

      1. *-commutative 72.6%

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

   5. Simplified 72.6%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+97}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+89}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -2.5e+89) x_m (if (<= z 1.12e+17) (/ 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 <= -2.5e+89) {
		tmp = x_m;
	} else if (z <= 1.12e+17) {
		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 <= (-2.5d+89)) then
        tmp = x_m
    else if (z <= 1.12d+17) 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 <= -2.5e+89) {
		tmp = x_m;
	} else if (z <= 1.12e+17) {
		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 <= -2.5e+89:
		tmp = x_m
	elif z <= 1.12e+17:
		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 <= -2.5e+89)
		tmp = x_m;
	elseif (z <= 1.12e+17)
		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 <= -2.5e+89)
		tmp = x_m;
	elseif (z <= 1.12e+17)
		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, -2.5e+89], x$95$m, If[LessEqual[z, 1.12e+17], N[(y / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.49999999999999992e89 or 1.12e17 < z

   1. Initial program 72.9%

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

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

      3. distribute-frac-neg2 99.9%

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

      4. neg-sub0 99.9%

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

      5. remove-double-neg 99.9%

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

      6. unsub-neg 99.9%

         \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

      7. div-sub 100.0%

         \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

      8. *-inverses 100.0%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

      9. metadata-eval 100.0%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

      10. associate--r- 100.0%

          \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. distribute-frac-neg2 100.0%

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

      13. remove-double-neg 100.0%

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

      14. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 79.8%

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

   if -2.49999999999999992e89 < z < 1.12e17

   1. Initial program 94.0%

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

      1. *-commutative 94.0%

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

      2. associate-/l* 90.9%

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

   3. Simplified 90.9%

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

      1. clear-num 90.3%

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

      2. un-div-inv 90.4%

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

   6. Applied egg-rr 90.4%

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

   7. Taylor expanded in y around inf 71.1%

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

Alternative 4: 71.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+89}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 10^{+17}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -2.8e+89) x_m (if (<= z 1e+17) (* 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 <= -2.8e+89) {
		tmp = x_m;
	} else if (z <= 1e+17) {
		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 <= (-2.8d+89)) then
        tmp = x_m
    else if (z <= 1d+17) 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 <= -2.8e+89) {
		tmp = x_m;
	} else if (z <= 1e+17) {
		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 <= -2.8e+89:
		tmp = x_m
	elif z <= 1e+17:
		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 <= -2.8e+89)
		tmp = x_m;
	elseif (z <= 1e+17)
		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 <= -2.8e+89)
		tmp = x_m;
	elseif (z <= 1e+17)
		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, -2.8e+89], x$95$m, If[LessEqual[z, 1e+17], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.7999999999999998e89 or 1e17 < z

   1. Initial program 72.9%

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

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

      3. distribute-frac-neg2 99.9%

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

      4. neg-sub0 99.9%

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

      5. remove-double-neg 99.9%

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

      6. unsub-neg 99.9%

         \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

      7. div-sub 100.0%

         \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

      8. *-inverses 100.0%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

      9. metadata-eval 100.0%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

      10. associate--r- 100.0%

          \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. distribute-frac-neg2 100.0%

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

      13. remove-double-neg 100.0%

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

      14. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 79.8%

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

   if -2.7999999999999998e89 < z < 1e17

   1. Initial program 94.0%

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

      1. associate-/l* 92.8%

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

      2. remove-double-neg 92.8%

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

      3. distribute-frac-neg2 92.8%

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

      4. neg-sub0 92.8%

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

      5. remove-double-neg 92.8%

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

      6. unsub-neg 92.8%

         \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

      7. div-sub 92.8%

         \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

      8. *-inverses 92.8%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

      9. metadata-eval 92.8%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

      10. associate--r- 92.8%

          \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

      11. neg-sub0 92.8%

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

      12. distribute-frac-neg2 92.8%

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

      13. remove-double-neg 92.8%

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

      14. sub-neg 92.8%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in y around inf 72.2%

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

      1. associate-*l/ 71.1%

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

      2. *-commutative 71.1%

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

   7. Simplified 71.1%

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

Alternative 5: 69.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+97}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+17}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -3.4e+97) x_m (if (<= z 1.15e+17) (* 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 <= -3.4e+97) {
		tmp = x_m;
	} else if (z <= 1.15e+17) {
		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 <= (-3.4d+97)) then
        tmp = x_m
    else if (z <= 1.15d+17) 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 <= -3.4e+97) {
		tmp = x_m;
	} else if (z <= 1.15e+17) {
		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 <= -3.4e+97:
		tmp = x_m
	elif z <= 1.15e+17:
		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 <= -3.4e+97)
		tmp = x_m;
	elseif (z <= 1.15e+17)
		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 <= -3.4e+97)
		tmp = x_m;
	elseif (z <= 1.15e+17)
		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, -3.4e+97], x$95$m, If[LessEqual[z, 1.15e+17], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.4000000000000001e97 or 1.15e17 < z

   1. Initial program 72.4%

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

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

      3. distribute-frac-neg2 100.0%

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

      4. neg-sub0 100.0%

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

      5. remove-double-neg 100.0%

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

      6. unsub-neg 100.0%

         \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

      7. div-sub 100.0%

         \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

      8. *-inverses 100.0%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

      9. metadata-eval 100.0%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

      10. associate--r- 100.0%

          \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. distribute-frac-neg2 100.0%

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

      13. remove-double-neg 100.0%

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

      14. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 81.2%

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

   if -3.4000000000000001e97 < z < 1.15e17

   1. Initial program 94.1%

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

      1. associate-/l* 92.9%

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

      2. remove-double-neg 92.9%

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

      3. distribute-frac-neg2 92.9%

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

      4. neg-sub0 92.9%

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

      5. remove-double-neg 92.9%

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

      6. unsub-neg 92.9%

         \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

      7. div-sub 92.9%

         \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

      8. *-inverses 92.9%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

      9. metadata-eval 92.9%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

      10. associate--r- 92.9%

          \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

      11. neg-sub0 92.9%

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

      12. distribute-frac-neg2 92.9%

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

      13. remove-double-neg 92.9%

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

      14. sub-neg 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in y around inf 69.0%

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

Alternative 6: 96.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.92 \cdot 10^{-191}:\\ \;\;\;\;x\_m + \frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m + x\_m \cdot \frac{y}{z}\\ \end{array} \end{array} \]

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 1.92e-191) (+ x_m (/ (* x_m y) 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 <= 1.92e-191) {
		tmp = x_m + ((x_m * y) / 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 <= 1.92d-191) then
        tmp = x_m + ((x_m * y) / 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 <= 1.92e-191) {
		tmp = x_m + ((x_m * y) / 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 <= 1.92e-191:
		tmp = x_m + ((x_m * y) / 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 <= 1.92e-191)
		tmp = Float64(x_m + Float64(Float64(x_m * y) / 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 <= 1.92e-191)
		tmp = x_m + ((x_m * y) / 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, 1.92e-191], N[(x$95$m + N[(N[(x$95$m * y), $MachinePrecision] / z), $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 < 1.9199999999999999e-191

   1. Initial program 89.7%

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

      1. associate-/l* 93.5%

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

      2. remove-double-neg 93.5%

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

      3. distribute-frac-neg2 93.5%

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

      4. neg-sub0 93.5%

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

      5. remove-double-neg 93.5%

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

      6. unsub-neg 93.5%

         \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

      7. div-sub 93.5%

         \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

      8. *-inverses 93.5%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

      9. metadata-eval 93.5%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

      10. associate--r- 93.5%

          \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

      11. neg-sub0 93.5%

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

      12. distribute-frac-neg2 93.5%

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

      13. remove-double-neg 93.5%

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

      14. sub-neg 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in y around 0 94.4%

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

   if 1.9199999999999999e-191 < x

   1. Initial program 78.8%

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

      1. associate-/l* 99.0%

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

      2. remove-double-neg 99.0%

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

      3. distribute-frac-neg2 99.0%

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

      4. neg-sub0 99.0%

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

      5. remove-double-neg 99.0%

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

      6. unsub-neg 99.0%

         \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

      7. div-sub 99.1%

         \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

      8. *-inverses 99.1%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

      9. metadata-eval 99.1%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

      10. associate--r- 99.1%

          \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

      11. neg-sub0 99.1%

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

      12. distribute-frac-neg2 99.1%

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

      13. remove-double-neg 99.1%

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

      14. sub-neg 99.1%

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

   3. Simplified 99.1%

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

      1. sub-neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. distribute-rgt-in 99.1%

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

      4. *-un-lft-identity 99.1%

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.92 \cdot 10^{-191}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.9 \cdot 10^{-191}:\\ \;\;\;\;x\_m + \frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - -1\right)\\ \end{array} \end{array} \]

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 (<= x_m 1.9e-191) (+ x_m (/ (* x_m y) 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 (x_m <= 1.9e-191) {
		tmp = x_m + ((x_m * y) / 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 (x_m <= 1.9d-191) then
        tmp = x_m + ((x_m * y) / 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 (x_m <= 1.9e-191) {
		tmp = x_m + ((x_m * y) / 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 x_m <= 1.9e-191:
		tmp = x_m + ((x_m * y) / 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 (x_m <= 1.9e-191)
		tmp = Float64(x_m + Float64(Float64(x_m * y) / 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 (x_m <= 1.9e-191)
		tmp = x_m + ((x_m * y) / 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[x$95$m, 1.9e-191], N[(x$95$m + N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999e-191

   1. Initial program 89.7%

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

      1. associate-/l* 93.5%

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

      2. remove-double-neg 93.5%

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

      3. distribute-frac-neg2 93.5%

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

      4. neg-sub0 93.5%

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

      5. remove-double-neg 93.5%

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

      6. unsub-neg 93.5%

         \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

      7. div-sub 93.5%

         \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

      8. *-inverses 93.5%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

      9. metadata-eval 93.5%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

      10. associate--r- 93.5%

          \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

      11. neg-sub0 93.5%

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

      12. distribute-frac-neg2 93.5%

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

      13. remove-double-neg 93.5%

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

      14. sub-neg 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in y around 0 94.4%

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

   if 1.8999999999999999e-191 < x

   1. Initial program 78.8%

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

      1. associate-/l* 99.0%

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

      2. remove-double-neg 99.0%

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

      3. distribute-frac-neg2 99.0%

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

      4. neg-sub0 99.0%

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

      5. remove-double-neg 99.0%

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

      6. unsub-neg 99.0%

         \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

      7. div-sub 99.1%

         \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

      8. *-inverses 99.1%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

      9. metadata-eval 99.1%

         \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

      10. associate--r- 99.1%

          \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

      11. neg-sub0 99.1%

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

      12. distribute-frac-neg2 99.1%

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

      13. remove-double-neg 99.1%

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

      14. sub-neg 99.1%

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

   3. Simplified 99.1%

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

Alternative 8: 95.8% 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 85.1%

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

   1. associate-/l* 95.8%

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

   2. remove-double-neg 95.8%

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

   3. distribute-frac-neg2 95.8%

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

   4. neg-sub0 95.8%

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

   5. remove-double-neg 95.8%

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

   6. unsub-neg 95.8%

      \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

   7. div-sub 95.8%

      \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

   8. *-inverses 95.8%

      \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

   9. metadata-eval 95.8%

      \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

   10. associate--r- 95.8%

       \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

   11. neg-sub0 95.8%

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

   12. distribute-frac-neg2 95.8%

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

   13. remove-double-neg 95.8%

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

   14. sub-neg 95.8%

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

3. Simplified 95.8%

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

Alternative 9: 50.2% 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 85.1%

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

   1. associate-/l* 95.8%

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

   2. remove-double-neg 95.8%

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

   3. distribute-frac-neg2 95.8%

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

   4. neg-sub0 95.8%

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

   5. remove-double-neg 95.8%

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

   6. unsub-neg 95.8%

      \[\leadsto x \cdot \left(0 - \frac{\color{blue}{y - \left(-z\right)}}{-z}\right) \]

   7. div-sub 95.8%

      \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{y}{-z} - \frac{-z}{-z}\right)}\right) \]

   8. *-inverses 95.8%

      \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{1}\right)\right) \]

   9. metadata-eval 95.8%

      \[\leadsto x \cdot \left(0 - \left(\frac{y}{-z} - \color{blue}{\left(--1\right)}\right)\right) \]

   10. associate--r- 95.8%

       \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{y}{-z}\right) + \left(--1\right)\right)} \]

   11. neg-sub0 95.8%

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

   12. distribute-frac-neg2 95.8%

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

   13. remove-double-neg 95.8%

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

   14. sub-neg 95.8%

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

3. Simplified 95.8%

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

5. Taylor expanded in y around 0 48.0%

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

Developer Target 1: 96.1% 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 2024146 
(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))