Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Percentage Accurate: 94.0% → 99.3%

Time: 15.5s

Alternatives: 21

Speedup: 1.0×

• 31.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

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 - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Initial Program: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

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 - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} + \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \frac{0.083333333333333}{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e-66)
   (+
    (/
     (+
      0.083333333333333
      (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z))
     x)
    (fma -0.5 (log x) 0.91893853320467))
   (+
    (fma
     (fma (/ y x) z (/ (fma 0.0007936500793651 z -0.0027777777777778) x))
     z
     (/ 0.083333333333333 x))
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e-66) {
		tmp = ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x) + fma(-0.5, log(x), 0.91893853320467);
	} else {
		tmp = fma(fma((y / x), z, (fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, (0.083333333333333 / x)) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e-66)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x) + fma(-0.5, log(x), 0.91893853320467));
	else
		tmp = Float64(fma(fma(Float64(y / x), z, Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, Float64(0.083333333333333 / x)) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e-66], N[(N[(N[(0.083333333333333 + N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / x), $MachinePrecision] * z + N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * z + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-66

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \log x + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lower-log.f64 99.7

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\log x}, 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 2e-66 < x

   1. Initial program 88.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \left(\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) \cdot z} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}, z, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

   5. Applied rewrites 99.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \frac{0.083333333333333}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} + \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \frac{0.083333333333333}{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) - \frac{\left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+83}:\\ \;\;\;\;\left(\frac{\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651}{x} \cdot z\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.0027777777777778 - \frac{0.083333333333333}{z}}{z} - y\right) - 0.0007936500793651\right) \cdot \left(\left(\frac{-1}{x} \cdot z\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (-
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (-
            (* (- 0.0027777777777778 (* z (+ 0.0007936500793651 y))) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -5e+83)
     (*
      (*
       (/
        (+
         (+ (/ (- (/ 0.083333333333333 z) 0.0027777777777778) z) y)
         0.0007936500793651)
        x)
       z)
      z)
     (if (<= t_0 5e+307)
       (fma
        (- x 0.5)
        (log x)
        (- (+ (/ 0.083333333333333 x) 0.91893853320467) x))
       (*
        (-
         (- (/ (- 0.0027777777777778 (/ 0.083333333333333 z)) z) y)
         0.0007936500793651)
        (* (* (/ -1.0 x) z) z))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) - ((((0.0027777777777778 - (z * (0.0007936500793651 + y))) * z) - 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -5e+83) {
		tmp = (((((((0.083333333333333 / z) - 0.0027777777777778) / z) + y) + 0.0007936500793651) / x) * z) * z;
	} else if (t_0 <= 5e+307) {
		tmp = fma((x - 0.5), log(x), (((0.083333333333333 / x) + 0.91893853320467) - x));
	} else {
		tmp = ((((0.0027777777777778 - (0.083333333333333 / z)) / z) - y) - 0.0007936500793651) * (((-1.0 / x) * z) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) - Float64(Float64(Float64(Float64(0.0027777777777778 - Float64(z * Float64(0.0007936500793651 + y))) * z) - 0.083333333333333) / x))
	tmp = 0.0
	if (t_0 <= -5e+83)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.083333333333333 / z) - 0.0027777777777778) / z) + y) + 0.0007936500793651) / x) * z) * z);
	elseif (t_0 <= 5e+307)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(0.083333333333333 / x) + 0.91893853320467) - x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.0027777777777778 - Float64(0.083333333333333 / z)) / z) - y) - 0.0007936500793651) * Float64(Float64(Float64(-1.0 / x) * z) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - N[(N[(N[(N[(0.0027777777777778 - N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+83], N[(N[(N[(N[(N[(N[(N[(N[(0.083333333333333 / z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / z), $MachinePrecision] + y), $MachinePrecision] + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 5e+307], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(0.083333333333333 / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0027777777777778 - N[(0.083333333333333 / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] - y), $MachinePrecision] - 0.0007936500793651), $MachinePrecision] * N[(N[(N[(-1.0 / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -5.00000000000000029e83

   1. Initial program 86.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 91.5%

      \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.6%

       \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \left(\frac{1}{-x} \cdot \left(\left(-z\right) \cdot \color{blue}{z}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 91.6%

       \[\leadsto \left(\frac{\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651}{-x} \cdot \left(-z\right)\right) \cdot z \]

   if -5.00000000000000029e83 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 5e307

   1. Initial program 99.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)} - x \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)\right)} - x \]

      3. associate--l+ N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right) + \left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} + \left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x - \frac{1}{2}}, \log x, \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right) \]

      7. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \color{blue}{\log x}, \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right) \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{91893853320467}{100000000000000}\right)} - x\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{91893853320467}{100000000000000}\right)} - x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}} + \frac{91893853320467}{100000000000000}\right) - x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) \]

      13. lower-/.f64 91.8

          \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(\color{blue}{\frac{0.083333333333333}{x}} + 0.91893853320467\right) - x\right) \]

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)} \]

   if 5e307 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 84.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.4%

      \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 89.6%

       \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \left(\frac{1}{-x} \cdot \left(\left(-z\right) \cdot \color{blue}{z}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 90.5%

       \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \left(\left(\frac{-1}{x} \cdot z\right) \cdot \left(-z\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) - \frac{\left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - 0.083333333333333}{x} \leq -5 \cdot 10^{+83}:\\ \;\;\;\;\left(\frac{\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651}{x} \cdot z\right) \cdot z\\ \mathbf{elif}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) - \frac{\left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - 0.083333333333333}{x} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.0027777777777778 - \frac{0.083333333333333}{z}}{z} - y\right) - 0.0007936500793651\right) \cdot \left(\left(\frac{-1}{x} \cdot z\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) - \frac{\left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+83}:\\ \;\;\;\;\left(\frac{\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651}{x} \cdot z\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) - \left(x - \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.0027777777777778 - \frac{0.083333333333333}{z}}{z} - y\right) - 0.0007936500793651\right) \cdot \left(\left(\frac{-1}{x} \cdot z\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (-
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (-
            (* (- 0.0027777777777778 (* z (+ 0.0007936500793651 y))) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -5e+83)
     (*
      (*
       (/
        (+
         (+ (/ (- (/ 0.083333333333333 z) 0.0027777777777778) z) y)
         0.0007936500793651)
        x)
       z)
      z)
     (if (<= t_0 5e+307)
       (-
        (fma (- x 0.5) (log x) 0.91893853320467)
        (- x (/ 0.083333333333333 x)))
       (*
        (-
         (- (/ (- 0.0027777777777778 (/ 0.083333333333333 z)) z) y)
         0.0007936500793651)
        (* (* (/ -1.0 x) z) z))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) - ((((0.0027777777777778 - (z * (0.0007936500793651 + y))) * z) - 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -5e+83) {
		tmp = (((((((0.083333333333333 / z) - 0.0027777777777778) / z) + y) + 0.0007936500793651) / x) * z) * z;
	} else if (t_0 <= 5e+307) {
		tmp = fma((x - 0.5), log(x), 0.91893853320467) - (x - (0.083333333333333 / x));
	} else {
		tmp = ((((0.0027777777777778 - (0.083333333333333 / z)) / z) - y) - 0.0007936500793651) * (((-1.0 / x) * z) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) - Float64(Float64(Float64(Float64(0.0027777777777778 - Float64(z * Float64(0.0007936500793651 + y))) * z) - 0.083333333333333) / x))
	tmp = 0.0
	if (t_0 <= -5e+83)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.083333333333333 / z) - 0.0027777777777778) / z) + y) + 0.0007936500793651) / x) * z) * z);
	elseif (t_0 <= 5e+307)
		tmp = Float64(fma(Float64(x - 0.5), log(x), 0.91893853320467) - Float64(x - Float64(0.083333333333333 / x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.0027777777777778 - Float64(0.083333333333333 / z)) / z) - y) - 0.0007936500793651) * Float64(Float64(Float64(-1.0 / x) * z) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - N[(N[(N[(N[(0.0027777777777778 - N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+83], N[(N[(N[(N[(N[(N[(N[(N[(0.083333333333333 / z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] / z), $MachinePrecision] + y), $MachinePrecision] + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 5e+307], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] - N[(x - N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0027777777777778 - N[(0.083333333333333 / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] - y), $MachinePrecision] - 0.0007936500793651), $MachinePrecision] * N[(N[(N[(-1.0 / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -5.00000000000000029e83

   1. Initial program 86.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 91.5%

      \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.6%

       \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \left(\frac{1}{-x} \cdot \left(\left(-z\right) \cdot \color{blue}{z}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 91.6%

       \[\leadsto \left(\frac{\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651}{-x} \cdot \left(-z\right)\right) \cdot z \]

   if -5.00000000000000029e83 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 5e307

   1. Initial program 99.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 3.5

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

   5. Applied rewrites 3.5%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) + \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) + \frac{91893853320467}{100000000000000}} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)} + \frac{91893853320467}{100000000000000} \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right)} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) \]

      9. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} - x\right)} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} - x\right)} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}} - x\right) + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} - x\right) + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} - x\right) + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} - x\right) + \left(\color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} + \frac{91893853320467}{100000000000000}\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} - x\right) + \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \frac{91893853320467}{100000000000000}\right)} \]

      16. lower--.f64 N/A

          \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} - x\right) + \mathsf{fma}\left(\color{blue}{x - \frac{1}{2}}, \log x, \frac{91893853320467}{100000000000000}\right) \]

      17. lower-log.f64 91.7

          \[\leadsto \left(\frac{0.083333333333333}{x} - x\right) + \mathsf{fma}\left(x - 0.5, \color{blue}{\log x}, 0.91893853320467\right) \]

   8. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\left(\frac{0.083333333333333}{x} - x\right) + \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right)} \]

   if 5e307 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 84.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.4%

      \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 89.6%

       \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \left(\frac{1}{-x} \cdot \left(\left(-z\right) \cdot \color{blue}{z}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 90.5%

       \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \left(\left(\frac{-1}{x} \cdot z\right) \cdot \left(-z\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) - \frac{\left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - 0.083333333333333}{x} \leq -5 \cdot 10^{+83}:\\ \;\;\;\;\left(\frac{\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651}{x} \cdot z\right) \cdot z\\ \mathbf{elif}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) - \frac{\left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - 0.083333333333333}{x} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) - \left(x - \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.0027777777777778 - \frac{0.083333333333333}{z}}{z} - y\right) - 0.0007936500793651\right) \cdot \left(\left(\frac{-1}{x} \cdot z\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) - \frac{\left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - 0.083333333333333}{x} \leq -5 \cdot 10^{+83}:\\ \;\;\;\;\left(\frac{y}{x} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (-
       (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
       (/
        (-
         (* (- 0.0027777777777778 (* z (+ 0.0007936500793651 y))) z)
         0.083333333333333)
        x))
      -5e+83)
   (* (* (/ y x) z) z)
   (/
    (fma (fma 0.0007936500793651 z -0.0027777777777778) z 0.083333333333333)
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((x - 0.5) * log(x)) - x) + 0.91893853320467) - ((((0.0027777777777778 - (z * (0.0007936500793651 + y))) * z) - 0.083333333333333) / x)) <= -5e+83) {
		tmp = ((y / x) * z) * z;
	} else {
		tmp = fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) - Float64(Float64(Float64(Float64(0.0027777777777778 - Float64(z * Float64(0.0007936500793651 + y))) * z) - 0.083333333333333) / x)) <= -5e+83)
		tmp = Float64(Float64(Float64(y / x) * z) * z);
	else
		tmp = Float64(fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - N[(N[(N[(N[(0.0027777777777778 - N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], -5e+83], N[(N[(N[(y / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -5.00000000000000029e83

   1. Initial program 86.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 85.7%

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

   if -5.00000000000000029e83 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 94.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right) + \frac{91893853320467}{100000000000000}} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) - \frac{\left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - 0.083333333333333}{x} \leq -5 \cdot 10^{+83}:\\ \;\;\;\;\left(\frac{y}{x} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} + \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e-66)
   (+
    (/
     (+
      0.083333333333333
      (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z))
     x)
    (fma -0.5 (log x) 0.91893853320467))
   (-
    (fma
     (fma (/ y x) z (/ (fma 0.0007936500793651 z -0.0027777777777778) x))
     z
     (fma (- x 0.5) (log x) (/ 0.083333333333333 x)))
    (- x 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e-66) {
		tmp = ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x) + fma(-0.5, log(x), 0.91893853320467);
	} else {
		tmp = fma(fma((y / x), z, (fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, fma((x - 0.5), log(x), (0.083333333333333 / x))) - (x - 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e-66)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x) + fma(-0.5, log(x), 0.91893853320467));
	else
		tmp = Float64(fma(fma(Float64(y / x), z, Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, fma(Float64(x - 0.5), log(x), Float64(0.083333333333333 / x))) - Float64(x - 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e-66], N[(N[(N[(0.083333333333333 + N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / x), $MachinePrecision] * z + N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * z + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-66

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \log x + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lower-log.f64 99.7

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\log x}, 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 2e-66 < x

   1. Initial program 88.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} + \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+34}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{0.5 + x} - x\right) + 0.91893853320467\right) - \frac{\left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\right) - \left(x - 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.2e+34)
   (-
    (+ (- (/ (* (fma x x -0.25) (log x)) (+ 0.5 x)) x) 0.91893853320467)
    (/
     (-
      (* (- 0.0027777777777778 (* z (+ 0.0007936500793651 y))) z)
      0.083333333333333)
     x))
   (-
    (fma (- x 0.5) (log x) (* (* (/ z x) (+ 0.0007936500793651 y)) z))
    (- x 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.2e+34) {
		tmp = ((((fma(x, x, -0.25) * log(x)) / (0.5 + x)) - x) + 0.91893853320467) - ((((0.0027777777777778 - (z * (0.0007936500793651 + y))) * z) - 0.083333333333333) / x);
	} else {
		tmp = fma((x - 0.5), log(x), (((z / x) * (0.0007936500793651 + y)) * z)) - (x - 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.2e+34)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(x, x, -0.25) * log(x)) / Float64(0.5 + x)) - x) + 0.91893853320467) - Float64(Float64(Float64(Float64(0.0027777777777778 - Float64(z * Float64(0.0007936500793651 + y))) * z) - 0.083333333333333) / x));
	else
		tmp = Float64(fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(z / x) * Float64(0.0007936500793651 + y)) * z)) - Float64(x - 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.2e+34], N[(N[(N[(N[(N[(N[(x * x + -0.25), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] / N[(0.5 + x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - N[(N[(N[(N[(0.0027777777777778 - N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999993e34

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(x - \frac{1}{2}\right)} \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{x \cdot x - \frac{1}{2} \cdot \frac{1}{2}}{x + \frac{1}{2}}} \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      4. associate-*l/ N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - \frac{1}{2} \cdot \frac{1}{2}\right) \cdot \log x}{x + \frac{1}{2}}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\left(x \cdot x - \frac{1}{2} \cdot \frac{1}{2}\right) \cdot \log x}{x + \frac{1}{2}}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x \cdot x - \frac{1}{2} \cdot \frac{1}{2}\right) \cdot \log x}}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      7. sub-neg N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x \cdot x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)\right)} \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)} \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right) \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{\frac{-1}{4}}\right) \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \frac{-1}{4}\right) \cdot \log x}{\color{blue}{\frac{1}{2} + x}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      12. lower-+.f64 99.7

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{\color{blue}{0.5 + x}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied rewrites 99.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{0.5 + x}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 1.19999999999999993e34 < x

   1. Initial program 85.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 12.0

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

   5. Applied rewrites 12.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right) - x} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(x\right)\right)\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(x\right)\right)\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)} \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} - x\right) + \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} - x\right) + \left(\color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   8. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\left(0.91893853320467 - x\right) + \mathsf{fma}\left(x - 0.5, \log x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(\frac{y}{x} \cdot z, z, \frac{0.083333333333333}{x}\right)\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\frac{91893853320467}{100000000000000} - x\right) + \mathsf{fma}\left(x - \frac{1}{2}, \log x, {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 99.5%

       \[\leadsto \left(0.91893853320467 - x\right) + \mathsf{fma}\left(x - 0.5, \log x, \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) \cdot z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+34}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{0.5 + x} - x\right) + 0.91893853320467\right) - \frac{\left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\right) - \left(x - 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+34}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) - \frac{\left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\right) - \left(x - 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.2e+34)
   (-
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/
     (-
      (* (- 0.0027777777777778 (* z (+ 0.0007936500793651 y))) z)
      0.083333333333333)
     x))
   (-
    (fma (- x 0.5) (log x) (* (* (/ z x) (+ 0.0007936500793651 y)) z))
    (- x 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.2e+34) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) - ((((0.0027777777777778 - (z * (0.0007936500793651 + y))) * z) - 0.083333333333333) / x);
	} else {
		tmp = fma((x - 0.5), log(x), (((z / x) * (0.0007936500793651 + y)) * z)) - (x - 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.2e+34)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) - Float64(Float64(Float64(Float64(0.0027777777777778 - Float64(z * Float64(0.0007936500793651 + y))) * z) - 0.083333333333333) / x));
	else
		tmp = Float64(fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(z / x) * Float64(0.0007936500793651 + y)) * z)) - Float64(x - 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.2e+34], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - N[(N[(N[(N[(0.0027777777777778 - N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999993e34

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   if 1.19999999999999993e34 < x

   1. Initial program 85.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 12.0

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

   5. Applied rewrites 12.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right) - x} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(x\right)\right)\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(x\right)\right)\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)} \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} - x\right) + \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} - x\right) + \left(\color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   8. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\left(0.91893853320467 - x\right) + \mathsf{fma}\left(x - 0.5, \log x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(\frac{y}{x} \cdot z, z, \frac{0.083333333333333}{x}\right)\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\frac{91893853320467}{100000000000000} - x\right) + \mathsf{fma}\left(x - \frac{1}{2}, \log x, {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 99.5%

       \[\leadsto \left(0.91893853320467 - x\right) + \mathsf{fma}\left(x - 0.5, \log x, \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) \cdot z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+34}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) - \frac{\left(0.0027777777777778 - z \cdot \left(0.0007936500793651 + y\right)\right) \cdot z - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\right) - \left(x - 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-20}:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} + \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\right) - \left(x - 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.1e-20)
   (+
    (/
     (+
      0.083333333333333
      (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z))
     x)
    (fma -0.5 (log x) 0.91893853320467))
   (-
    (fma (- x 0.5) (log x) (* (* (/ z x) (+ 0.0007936500793651 y)) z))
    (- x 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.1e-20) {
		tmp = ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x) + fma(-0.5, log(x), 0.91893853320467);
	} else {
		tmp = fma((x - 0.5), log(x), (((z / x) * (0.0007936500793651 + y)) * z)) - (x - 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.1e-20)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x) + fma(-0.5, log(x), 0.91893853320467));
	else
		tmp = Float64(fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(z / x) * Float64(0.0007936500793651 + y)) * z)) - Float64(x - 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.1e-20], N[(N[(N[(0.083333333333333 + N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-20

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \log x + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lower-log.f64 99.7

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\log x}, 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 3.1e-20 < x

   1. Initial program 86.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 15.0

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

   5. Applied rewrites 15.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right) - x} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(x\right)\right)\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(x\right)\right)\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)} \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} - x\right) + \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} - x\right) + \left(\color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   8. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\left(0.91893853320467 - x\right) + \mathsf{fma}\left(x - 0.5, \log x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(\frac{y}{x} \cdot z, z, \frac{0.083333333333333}{x}\right)\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\frac{91893853320467}{100000000000000} - x\right) + \mathsf{fma}\left(x - \frac{1}{2}, \log x, {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 98.9%

       \[\leadsto \left(0.91893853320467 - x\right) + \mathsf{fma}\left(x - 0.5, \log x, \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) \cdot z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-20}:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} + \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\right) - \left(x - 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\right) - \left(x - 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.1e-20)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (-
    (fma (- x 0.5) (log x) (* (* (/ z x) (+ 0.0007936500793651 y)) z))
    (- x 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.1e-20) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = fma((x - 0.5), log(x), (((z / x) * (0.0007936500793651 + y)) * z)) - (x - 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.1e-20)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(z / x) * Float64(0.0007936500793651 + y)) * z)) - Float64(x - 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.1e-20], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-20

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y + \frac{7936500793651}{10000000000000000}}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      10. lower-+.f64 99.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y + 0.0007936500793651}, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}} \]

   if 3.1e-20 < x

   1. Initial program 86.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 15.0

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

   5. Applied rewrites 15.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right) - x} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(x\right)\right)\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(x\right)\right)\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right)} \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} - x\right) + \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} - x\right) + \left(\color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   8. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\left(0.91893853320467 - x\right) + \mathsf{fma}\left(x - 0.5, \log x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(\frac{y}{x} \cdot z, z, \frac{0.083333333333333}{x}\right)\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\frac{91893853320467}{100000000000000} - x\right) + \mathsf{fma}\left(x - \frac{1}{2}, \log x, {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 98.9%

       \[\leadsto \left(0.91893853320467 - x\right) + \mathsf{fma}\left(x - 0.5, \log x, \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) \cdot z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\right) - \left(x - 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \log x \cdot x\right) - \left(x - 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.5)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (-
    (fma (/ (fma 0.0007936500793651 z -0.0027777777777778) x) z (* (log x) x))
    (- x 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.5) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = fma((fma(0.0007936500793651, z, -0.0027777777777778) / x), z, (log(x) * x)) - (x - 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.5)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(fma(Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x), z, Float64(log(x) * x)) - Float64(x - 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.5], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y + \frac{7936500793651}{10000000000000000}}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      10. lower-+.f64 99.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y + 0.0007936500793651}, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}} \]

   if 6.5 < x

   1. Initial program 86.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right) + \frac{91893853320467}{100000000000000}} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

   5. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right)}{x}, z, -1 \cdot \left(x \cdot \log \left(\frac{1}{x}\right)\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \log x \cdot x\right) - \left(x - 0.91893853320467\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \log x \cdot x\right) - \left(x - 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.2e+15)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.2e+15) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.2e+15)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.2e+15], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2e15

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y + \frac{7936500793651}{10000000000000000}}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      10. lower-+.f64 99.1

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y + 0.0007936500793651}, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}} \]

   if 1.2e15 < x

   1. Initial program 86.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) - 1\right) \cdot x \]

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 78.2

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

   5. Applied rewrites 78.2%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\left(\frac{y}{x} \cdot z\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          0.083333333333333
          (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z))))
   (if (<= t_0 -1.0)
     (* (* (/ y x) z) z)
     (if (<= t_0 0.1)
       (/ 1.0 (* 12.000000000000048 x))
       (* (* (/ z x) z) 0.0007936500793651)))))

C

double code(double x, double y, double z) {
	double t_0 = 0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = ((y / x) * z) * z;
	} else if (t_0 <= 0.1) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.083333333333333d0 + (((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0) * z)
    if (t_0 <= (-1.0d0)) then
        tmp = ((y / x) * z) * z
    else if (t_0 <= 0.1d0) then
        tmp = 1.0d0 / (12.000000000000048d0 * x)
    else
        tmp = ((z / x) * z) * 0.0007936500793651d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = ((y / x) * z) * z;
	} else if (t_0 <= 0.1) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)
	tmp = 0
	if t_0 <= -1.0:
		tmp = ((y / x) * z) * z
	elif t_0 <= 0.1:
		tmp = 1.0 / (12.000000000000048 * x)
	else:
		tmp = ((z / x) * z) * 0.0007936500793651
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.083333333333333 + Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z))
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = Float64(Float64(Float64(y / x) * z) * z);
	elseif (t_0 <= 0.1)
		tmp = Float64(1.0 / Float64(12.000000000000048 * x));
	else
		tmp = Float64(Float64(Float64(z / x) * z) * 0.0007936500793651);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z);
	tmp = 0.0;
	if (t_0 <= -1.0)
		tmp = ((y / x) * z) * z;
	elseif (t_0 <= 0.1)
		tmp = 1.0 / (12.000000000000048 * x);
	else
		tmp = ((z / x) * z) * 0.0007936500793651;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.083333333333333 + N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[(N[(N[(y / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * 0.0007936500793651), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -1

   1. Initial program 86.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 71.8%

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

   if -1 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

   5. Applied rewrites 18.4%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 23.1%

      \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.0%

       \[\leadsto \frac{0.083333333333333}{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 47.1%

       \[\leadsto \frac{1}{12.000000000000048 \cdot x} \]

   if 0.10000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 89.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right) + \frac{91893853320467}{100000000000000}} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

   5. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(x - 0.5, \log x, \frac{1}{x \cdot 12.000000000000048}\right)\right) - \left(x - 0.91893853320467\right) \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{{z}^{2}}{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 68.8%

       \[\leadsto \left(z \cdot \frac{z}{x}\right) \cdot \color{blue}{0.0007936500793651} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z \leq -1:\\ \;\;\;\;\left(\frac{y}{x} \cdot z\right) \cdot z\\ \mathbf{elif}\;0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z \leq 0.1:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z\\ t_1 := \left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z))
        (t_1 (* (* (/ z x) z) (+ 0.0007936500793651 y))))
   (if (<= t_0 -1.0)
     t_1
     (if (<= t_0 5e+59)
       (/
        (fma
         (fma 0.0007936500793651 z -0.0027777777777778)
         z
         0.083333333333333)
        x)
       t_1))))

C

double code(double x, double y, double z) {
	double t_0 = ((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z;
	double t_1 = ((z / x) * z) * (0.0007936500793651 + y);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+59) {
		tmp = fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z)
	t_1 = Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y))
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = t_1;
	elseif (t_0 <= 5e+59)
		tmp = Float64(fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], t$95$1, If[LessEqual[t$95$0, 5e+59], N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -1 or 4.9999999999999997e59 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 87.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

   5. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 79.6%

      \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\frac{7936500793651}{10000000000000000} + y\right) \cdot \left(z \cdot \frac{\color{blue}{z}}{x}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 78.4%

       \[\leadsto \left(y + 0.0007936500793651\right) \cdot \left(z \cdot \frac{\color{blue}{z}}{x}\right) \]

   if -1 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 4.9999999999999997e59

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right) + \frac{91893853320467}{100000000000000}} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z \leq -1:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \mathbf{elif}\;\left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\frac{z \cdot z}{x} \cdot y\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z)))
   (if (<= t_0 -1.0)
     (* (/ (* z z) x) y)
     (if (<= t_0 0.01)
       (/ 1.0 (* 12.000000000000048 x))
       (* (* (/ z x) z) 0.0007936500793651)))))

C

double code(double x, double y, double z) {
	double t_0 = ((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -1.0) {
		tmp = ((z * z) / x) * y;
	} else if (t_0 <= 0.01) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0) * z
    if (t_0 <= (-1.0d0)) then
        tmp = ((z * z) / x) * y
    else if (t_0 <= 0.01d0) then
        tmp = 1.0d0 / (12.000000000000048d0 * x)
    else
        tmp = ((z / x) * z) * 0.0007936500793651d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -1.0) {
		tmp = ((z * z) / x) * y;
	} else if (t_0 <= 0.01) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z
	tmp = 0
	if t_0 <= -1.0:
		tmp = ((z * z) / x) * y
	elif t_0 <= 0.01:
		tmp = 1.0 / (12.000000000000048 * x)
	else:
		tmp = ((z / x) * z) * 0.0007936500793651
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = Float64(Float64(Float64(z * z) / x) * y);
	elseif (t_0 <= 0.01)
		tmp = Float64(1.0 / Float64(12.000000000000048 * x));
	else
		tmp = Float64(Float64(Float64(z / x) * z) * 0.0007936500793651);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z;
	tmp = 0.0;
	if (t_0 <= -1.0)
		tmp = ((z * z) / x) * y;
	elseif (t_0 <= 0.01)
		tmp = 1.0 / (12.000000000000048 * x);
	else
		tmp = ((z / x) * z) * 0.0007936500793651;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[(N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * 0.0007936500793651), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -1

   1. Initial program 86.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 71.8%

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

   if -1 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 0.0100000000000000002

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

   5. Applied rewrites 18.4%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 23.1%

      \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.0%

       \[\leadsto \frac{0.083333333333333}{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 47.1%

       \[\leadsto \frac{1}{12.000000000000048 \cdot x} \]

   if 0.0100000000000000002 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 89.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right) + \frac{91893853320467}{100000000000000}} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

   5. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(x - 0.5, \log x, \frac{1}{x \cdot 12.000000000000048}\right)\right) - \left(x - 0.91893853320467\right) \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{{z}^{2}}{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 68.8%

       \[\leadsto \left(z \cdot \frac{z}{x}\right) \cdot \color{blue}{0.0007936500793651} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z \leq -1:\\ \;\;\;\;\frac{z \cdot z}{x} \cdot y\\ \mathbf{elif}\;\left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z \leq 0.01:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, 0.0007936500793651 + y, \frac{-0.0027777777777778}{x}\right), z, \frac{0.083333333333333}{x}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma
  (fma (/ z x) (+ 0.0007936500793651 y) (/ -0.0027777777777778 x))
  z
  (/ 0.083333333333333 x)))

C

double code(double x, double y, double z) {
	return fma(fma((z / x), (0.0007936500793651 + y), (-0.0027777777777778 / x)), z, (0.083333333333333 / x));
}

Julia

function code(x, y, z)
	return fma(fma(Float64(z / x), Float64(0.0007936500793651 + y), Float64(-0.0027777777777778 / x)), z, Float64(0.083333333333333 / x))
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision] + N[(-0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] * z + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.3%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

5. Applied rewrites 60.1%

   \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
7. Step-by-step derivation

8. Applied rewrites 52.9%

   \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]

9. Taylor expanded in z around 0

   \[\leadsto z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{83333333333333}{1000000000000000} \cdot \color{blue}{\frac{1}{x}} \]
10. Step-by-step derivation

11. Applied rewrites 63.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, y + 0.0007936500793651, \frac{-0.0027777777777778}{x}\right), z, \frac{0.083333333333333}{x}\right) \]

12. Final simplification 63.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, 0.0007936500793651 + y, \frac{-0.0027777777777778}{x}\right), z, \frac{0.083333333333333}{x}\right) \]
13. Add Preprocessing

Alternative 16: 46.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z \leq 0.1:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       0.083333333333333
       (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z))
      0.1)
   (/ 1.0 (* 12.000000000000048 x))
   (* (* (/ z x) z) 0.0007936500793651)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) <= 0.1) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((0.083333333333333d0 + (((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0) * z)) <= 0.1d0) then
        tmp = 1.0d0 / (12.000000000000048d0 * x)
    else
        tmp = ((z / x) * z) * 0.0007936500793651d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) <= 0.1) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) <= 0.1:
		tmp = 1.0 / (12.000000000000048 * x)
	else:
		tmp = ((z / x) * z) * 0.0007936500793651
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(0.083333333333333 + Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z)) <= 0.1)
		tmp = Float64(1.0 / Float64(12.000000000000048 * x));
	else
		tmp = Float64(Float64(Float64(z / x) * z) * 0.0007936500793651);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) <= 0.1)
		tmp = 1.0 / (12.000000000000048 * x);
	else
		tmp = ((z / x) * z) * 0.0007936500793651;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(0.083333333333333 + N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], 0.1], N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * 0.0007936500793651), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 95.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

   5. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.8%

      \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 34.4%

       \[\leadsto \frac{0.083333333333333}{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 34.5%

       \[\leadsto \frac{1}{12.000000000000048 \cdot x} \]

   if 0.10000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 89.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right) + \frac{91893853320467}{100000000000000}} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

   5. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(x - 0.5, \log x, \frac{1}{x \cdot 12.000000000000048}\right)\right) - \left(x - 0.91893853320467\right) \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{{z}^{2}}{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 68.8%

       \[\leadsto \left(z \cdot \frac{z}{x}\right) \cdot \color{blue}{0.0007936500793651} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z \leq 0.1:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z \leq 0.1:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       0.083333333333333
       (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z))
      0.1)
   (/ 1.0 (* 12.000000000000048 x))
   (* (/ 0.0007936500793651 x) (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) <= 0.1) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = (0.0007936500793651 / x) * (z * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((0.083333333333333d0 + (((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0) * z)) <= 0.1d0) then
        tmp = 1.0d0 / (12.000000000000048d0 * x)
    else
        tmp = (0.0007936500793651d0 / x) * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) <= 0.1) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = (0.0007936500793651 / x) * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) <= 0.1:
		tmp = 1.0 / (12.000000000000048 * x)
	else:
		tmp = (0.0007936500793651 / x) * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(0.083333333333333 + Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z)) <= 0.1)
		tmp = Float64(1.0 / Float64(12.000000000000048 * x));
	else
		tmp = Float64(Float64(0.0007936500793651 / x) * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) <= 0.1)
		tmp = 1.0 / (12.000000000000048 * x);
	else
		tmp = (0.0007936500793651 / x) * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(0.083333333333333 + N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], 0.1], N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.0007936500793651 / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 0.10000000000000001

   1. Initial program 95.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

   5. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.8%

      \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 34.4%

       \[\leadsto \frac{0.083333333333333}{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 34.5%

       \[\leadsto \frac{1}{12.000000000000048 \cdot x} \]

   if 0.10000000000000001 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 89.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right) + \frac{91893853320467}{100000000000000}} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

   5. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.5%

      \[\leadsto \left(\frac{0.0007936500793651}{x} - \frac{\frac{0.0027777777777778}{x}}{z}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{7936500793651}{10000000000000000}}{x} \cdot \left(z \cdot z\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 68.4%

       \[\leadsto \frac{0.0007936500793651}{x} \cdot \left(z \cdot z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z \leq 0.1:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.95e+105)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (* (/ z x) z) (+ 0.0007936500793651 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.95e+105) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((z / x) * z) * (0.0007936500793651 + y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.95e+105)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.95e+105], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.94999999999999989e105

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y + \frac{7936500793651}{10000000000000000}}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      10. lower-+.f64 84.1

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y + 0.0007936500793651}, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

   5. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}} \]

   if 1.94999999999999989e105 < x

   1. Initial program 78.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

   5. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 18.0%

      \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\frac{7936500793651}{10000000000000000} + y\right) \cdot \left(z \cdot \frac{\color{blue}{z}}{x}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 18.5%

       \[\leadsto \left(y + 0.0007936500793651\right) \cdot \left(z \cdot \frac{\color{blue}{z}}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 28.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -30:\\ \;\;\;\;\frac{z}{x} \cdot -0.0027777777777778\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -30.0)
   (* (/ z x) -0.0027777777777778)
   (/ 1.0 (* 12.000000000000048 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -30.0) {
		tmp = (z / x) * -0.0027777777777778;
	} else {
		tmp = 1.0 / (12.000000000000048 * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-30.0d0)) then
        tmp = (z / x) * (-0.0027777777777778d0)
    else
        tmp = 1.0d0 / (12.000000000000048d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -30.0) {
		tmp = (z / x) * -0.0027777777777778;
	} else {
		tmp = 1.0 / (12.000000000000048 * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -30.0:
		tmp = (z / x) * -0.0027777777777778
	else:
		tmp = 1.0 / (12.000000000000048 * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -30.0)
		tmp = Float64(Float64(z / x) * -0.0027777777777778);
	else
		tmp = Float64(1.0 / Float64(12.000000000000048 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -30.0)
		tmp = (z / x) * -0.0027777777777778;
	else
		tmp = 1.0 / (12.000000000000048 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -30.0], N[(N[(z / x), $MachinePrecision] * -0.0027777777777778), $MachinePrecision], N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -30

   1. Initial program 90.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - x\right) + \frac{91893853320467}{100000000000000}} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right)} \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.7%

      \[\leadsto \left(\frac{0.0007936500793651}{x} - \frac{\frac{0.0027777777777778}{x}}{z}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{-13888888888889}{5000000000000000} \cdot \frac{z}{\color{blue}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 21.6%

       \[\leadsto \frac{z}{x} \cdot -0.0027777777777778 \]

   if -30 < z

   1. Initial program 94.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

   5. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.1%

      \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.0%

       \[\leadsto \frac{0.083333333333333}{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 29.1%

       \[\leadsto \frac{1}{12.000000000000048 \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 23.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{12.000000000000048 \cdot x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 1.0 (* 12.000000000000048 x)))

C

double code(double x, double y, double z) {
	return 1.0 / (12.000000000000048 * x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 / (12.000000000000048d0 * x)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 / (12.000000000000048 * x);
}

Python

def code(x, y, z):
	return 1.0 / (12.000000000000048 * x)

Julia

function code(x, y, z)
	return Float64(1.0 / Float64(12.000000000000048 * x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 / (12.000000000000048 * x);
end

Wolfram

code[x_, y_, z_] := N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.3%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

5. Applied rewrites 60.1%

   \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
7. Step-by-step derivation

8. Applied rewrites 52.9%

   \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]

9. Taylor expanded in z around 0

   \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
10. Step-by-step derivation

11. Applied rewrites 23.1%

    \[\leadsto \frac{0.083333333333333}{x} \]
12. Step-by-step derivation

13. Applied rewrites 23.1%

    \[\leadsto \frac{1}{12.000000000000048 \cdot x} \]
14. Add Preprocessing

Alternative 21: 23.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 0.083333333333333 x))

C

double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.083333333333333d0 / x
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Python

def code(x, y, z):
	return 0.083333333333333 / x

Julia

function code(x, y, z)
	return Float64(0.083333333333333 / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 / x;
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 / x), $MachinePrecision]

Derivation

1. Initial program 93.3%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \left(\frac{\frac{91893853320467}{100000000000000}}{{z}^{2}} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x \cdot {z}^{2}} + \left(\frac{y}{x} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{{z}^{2}}\right)\right)\right)\right) - \left(\frac{\frac{13888888888889}{5000000000000000}}{x \cdot z} + \frac{x}{{z}^{2}}\right)\right)} \]

5. Applied rewrites 60.1%

   \[\leadsto \color{blue}{\left(\left(\left(\frac{0.91893853320467}{z \cdot z} + \mathsf{fma}\left(\frac{x - 0.5}{z}, \frac{\log x}{z}, \frac{y}{x}\right)\right) + \frac{\frac{1}{z}}{x} \cdot \left(\frac{0.083333333333333}{z} - 0.0027777777777778\right)\right) - \left(\frac{x}{z \cdot z} - \frac{0.0007936500793651}{x}\right)\right) \cdot \left(z \cdot z\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{{z}^{2}}\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{\color{blue}{x}} \]
7. Step-by-step derivation

8. Applied rewrites 52.9%

   \[\leadsto \left(\left(\frac{\frac{0.083333333333333}{z} - 0.0027777777777778}{z} + y\right) + 0.0007936500793651\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]

9. Taylor expanded in z around 0

   \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
10. Step-by-step derivation

11. Applied rewrites 23.1%

    \[\leadsto \frac{0.083333333333333}{x} \]
12. Add Preprocessing

Developer Target 1: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467 - x\right)\right) + \frac{0.083333333333333}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x))
  (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
}

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 - 0.5d0) * log(x)) + (0.91893853320467d0 - x)) + (0.083333333333333d0 / x)) + ((z / x) * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) + Float64(0.91893853320467 - x)) + Float64(0.083333333333333 / x)) + Float64(Float64(z / x) * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024249 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (* (- x 1/2) (log x)) (- 91893853320467/100000000000000 x)) (/ 83333333333333/1000000000000000 x)) (* (/ z x) (- (* z (+ y 7936500793651/10000000000000000)) 13888888888889/5000000000000000))))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))