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

Percentage Accurate: 93.9% → 99.4%

Time: 14.5s

Alternatives: 20

Speedup: 1.0×

• 31.6% 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: 93.9% 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: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{y}{x}, \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x}\right), \frac{0.083333333333333}{x}\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4e-56)
   (fma
    (fma
     z
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     0.083333333333333)
    (/ 1.0 x)
    (fma -0.5 (log x) 0.91893853320467))
   (+
    0.91893853320467
    (fma
     (log x)
     (+ x -0.5)
     (-
      (fma
       z
       (fma z (/ y x) (/ (fma z 0.0007936500793651 -0.0027777777777778) x))
       (/ 0.083333333333333 x))
      x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4e-56) {
		tmp = fma(fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), (1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	} else {
		tmp = 0.91893853320467 + fma(log(x), (x + -0.5), (fma(z, fma(z, (y / x), (fma(z, 0.0007936500793651, -0.0027777777777778) / x)), (0.083333333333333 / x)) - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4e-56)
		tmp = fma(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), Float64(1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	else
		tmp = Float64(0.91893853320467 + fma(log(x), Float64(x + -0.5), Float64(fma(z, fma(z, Float64(y / x), Float64(fma(z, 0.0007936500793651, -0.0027777777777778) / x)), Float64(0.083333333333333 / x)) - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4e-56], N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(N[(z * N[(z * N[(y / x), $MachinePrecision] + N[(N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0000000000000002e-56

   1. Initial program 99.8%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. lower-log.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 4.0000000000000002e-56 < x

   1. Initial program 88.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{y}{x}, \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x}\right), \frac{0.083333333333333}{x}\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := z \cdot \left(t\_0 - 0.0027777777777778\right)\\ t_2 := \mathsf{fma}\left(t\_0, z \cdot \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+74}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651)))
        (t_1 (* z (- t_0 0.0027777777777778)))
        (t_2 (fma t_0 (* z (/ 1.0 x)) (fma -0.5 (log x) 0.91893853320467))))
   (if (<= t_1 -2e+74)
     t_2
     (if (<= t_1 1e+68)
       (fma
        (/ 1.0 x)
        0.083333333333333
        (fma (log x) (+ x -0.5) (- 0.91893853320467 x)))
       t_2))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double t_1 = z * (t_0 - 0.0027777777777778);
	double t_2 = fma(t_0, (z * (1.0 / x)), fma(-0.5, log(x), 0.91893853320467));
	double tmp;
	if (t_1 <= -2e+74) {
		tmp = t_2;
	} else if (t_1 <= 1e+68) {
		tmp = fma((1.0 / x), 0.083333333333333, fma(log(x), (x + -0.5), (0.91893853320467 - x)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	t_1 = Float64(z * Float64(t_0 - 0.0027777777777778))
	t_2 = fma(t_0, Float64(z * Float64(1.0 / x)), fma(-0.5, log(x), 0.91893853320467))
	tmp = 0.0
	if (t_1 <= -2e+74)
		tmp = t_2;
	elseif (t_1 <= 1e+68)
		tmp = fma(Float64(1.0 / x), 0.083333333333333, fma(log(x), Float64(x + -0.5), Float64(0.91893853320467 - x)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(t$95$0 - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(z * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+74], t$95$2, If[LessEqual[t$95$1, 1e+68], N[(N[(1.0 / x), $MachinePrecision] * 0.083333333333333 + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

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) < -1.9999999999999999e74 or 9.99999999999999953e67 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 86.9%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 97.8

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

   5. Applied rewrites 97.8%

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 86.5

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

   10. Applied rewrites 86.5%

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

   if -1.9999999999999999e74 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 9.99999999999999953e67

   1. Initial program 99.5%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 95.9

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

   5. Applied rewrites 95.9%

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

   7. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467 - x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y + 0.0007936500793651\right), z \cdot \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y + 0.0007936500793651\right), z \cdot \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := z \cdot \left(t\_0 - 0.0027777777777778\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_1 \leq 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651)))
        (t_1 (* z (- t_0 0.0027777777777778))))
   (if (<= t_1 -2e+74)
     (* (* z y) (/ z x))
     (if (<= t_1 1e+68)
       (fma
        (/ 1.0 x)
        0.083333333333333
        (fma (log x) (+ x -0.5) (- 0.91893853320467 x)))
       (* z (/ t_0 x))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double t_1 = z * (t_0 - 0.0027777777777778);
	double tmp;
	if (t_1 <= -2e+74) {
		tmp = (z * y) * (z / x);
	} else if (t_1 <= 1e+68) {
		tmp = fma((1.0 / x), 0.083333333333333, fma(log(x), (x + -0.5), (0.91893853320467 - x)));
	} else {
		tmp = z * (t_0 / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	t_1 = Float64(z * Float64(t_0 - 0.0027777777777778))
	tmp = 0.0
	if (t_1 <= -2e+74)
		tmp = Float64(Float64(z * y) * Float64(z / x));
	elseif (t_1 <= 1e+68)
		tmp = fma(Float64(1.0 / x), 0.083333333333333, fma(log(x), Float64(x + -0.5), Float64(0.91893853320467 - x)));
	else
		tmp = Float64(z * Float64(t_0 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(t$95$0 - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+74], N[(N[(z * y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+68], N[(N[(1.0 / x), $MachinePrecision] * 0.083333333333333 + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(t$95$0 / x), $MachinePrecision]), $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) < -1.9999999999999999e74

   1. Initial program 91.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.9

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

   7. Applied rewrites 93.9%

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

   if -1.9999999999999999e74 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 9.99999999999999953e67

   1. Initial program 99.5%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 95.9

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

   5. Applied rewrites 95.9%

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

   7. Applied rewrites 96.1%

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

   if 9.99999999999999953e67 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 85.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 83.4

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

   8. Applied rewrites 83.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := z \cdot \left(t\_0 - 0.0027777777777778\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_1 \leq 10^{+68}:\\ \;\;\;\;0.91893853320467 + \left(\mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651)))
        (t_1 (* z (- t_0 0.0027777777777778))))
   (if (<= t_1 -2e+74)
     (* (* z y) (/ z x))
     (if (<= t_1 1e+68)
       (+
        0.91893853320467
        (- (fma (log x) (+ x -0.5) (/ 0.083333333333333 x)) x))
       (* z (/ t_0 x))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double t_1 = z * (t_0 - 0.0027777777777778);
	double tmp;
	if (t_1 <= -2e+74) {
		tmp = (z * y) * (z / x);
	} else if (t_1 <= 1e+68) {
		tmp = 0.91893853320467 + (fma(log(x), (x + -0.5), (0.083333333333333 / x)) - x);
	} else {
		tmp = z * (t_0 / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	t_1 = Float64(z * Float64(t_0 - 0.0027777777777778))
	tmp = 0.0
	if (t_1 <= -2e+74)
		tmp = Float64(Float64(z * y) * Float64(z / x));
	elseif (t_1 <= 1e+68)
		tmp = Float64(0.91893853320467 + Float64(fma(log(x), Float64(x + -0.5), Float64(0.083333333333333 / x)) - x));
	else
		tmp = Float64(z * Float64(t_0 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(t$95$0 - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+74], N[(N[(z * y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+68], N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(z * N[(t$95$0 / x), $MachinePrecision]), $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) < -1.9999999999999999e74

   1. Initial program 91.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.9

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

   7. Applied rewrites 93.9%

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

   if -1.9999999999999999e74 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 9.99999999999999953e67

   1. Initial program 99.5%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 95.9

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

   5. Applied rewrites 95.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-neg.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. associate-+l- N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 95.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 95.9

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

   7. Applied rewrites 95.9%

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

   if 9.99999999999999953e67 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 85.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 83.4

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

   8. Applied rewrites 83.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 10^{+68}:\\ \;\;\;\;0.91893853320467 + \left(\mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := z \cdot \left(t\_0 - 0.0027777777777778\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_1 \leq 10^{+68}:\\ \;\;\;\;\left(0.91893853320467 - x\right) + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651)))
        (t_1 (* z (- t_0 0.0027777777777778))))
   (if (<= t_1 -2e+74)
     (* (* z y) (/ z x))
     (if (<= t_1 1e+68)
       (+
        (- 0.91893853320467 x)
        (fma (log x) (+ x -0.5) (/ 0.083333333333333 x)))
       (* z (/ t_0 x))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double t_1 = z * (t_0 - 0.0027777777777778);
	double tmp;
	if (t_1 <= -2e+74) {
		tmp = (z * y) * (z / x);
	} else if (t_1 <= 1e+68) {
		tmp = (0.91893853320467 - x) + fma(log(x), (x + -0.5), (0.083333333333333 / x));
	} else {
		tmp = z * (t_0 / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	t_1 = Float64(z * Float64(t_0 - 0.0027777777777778))
	tmp = 0.0
	if (t_1 <= -2e+74)
		tmp = Float64(Float64(z * y) * Float64(z / x));
	elseif (t_1 <= 1e+68)
		tmp = Float64(Float64(0.91893853320467 - x) + fma(log(x), Float64(x + -0.5), Float64(0.083333333333333 / x)));
	else
		tmp = Float64(z * Float64(t_0 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(t$95$0 - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+74], N[(N[(z * y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+68], N[(N[(0.91893853320467 - x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(t$95$0 / x), $MachinePrecision]), $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) < -1.9999999999999999e74

   1. Initial program 91.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.9

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

   7. Applied rewrites 93.9%

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

   if -1.9999999999999999e74 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 9.99999999999999953e67

   1. Initial program 99.5%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if 9.99999999999999953e67 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 85.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 83.4

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

   8. Applied rewrites 83.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 10^{+68}:\\ \;\;\;\;\left(0.91893853320467 - x\right) + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := z \cdot \left(t\_0 - 0.0027777777777778\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_1 \leq 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right) - x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651)))
        (t_1 (* z (- t_0 0.0027777777777778))))
   (if (<= t_1 -2e+74)
     (* (* z y) (/ z x))
     (if (<= t_1 1e+68)
       (- (fma (log x) (+ x -0.5) (/ 0.083333333333333 x)) x)
       (* z (/ t_0 x))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double t_1 = z * (t_0 - 0.0027777777777778);
	double tmp;
	if (t_1 <= -2e+74) {
		tmp = (z * y) * (z / x);
	} else if (t_1 <= 1e+68) {
		tmp = fma(log(x), (x + -0.5), (0.083333333333333 / x)) - x;
	} else {
		tmp = z * (t_0 / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	t_1 = Float64(z * Float64(t_0 - 0.0027777777777778))
	tmp = 0.0
	if (t_1 <= -2e+74)
		tmp = Float64(Float64(z * y) * Float64(z / x));
	elseif (t_1 <= 1e+68)
		tmp = Float64(fma(log(x), Float64(x + -0.5), Float64(0.083333333333333 / x)) - x);
	else
		tmp = Float64(z * Float64(t_0 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(t$95$0 - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+74], N[(N[(z * y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+68], N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(z * N[(t$95$0 / x), $MachinePrecision]), $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) < -1.9999999999999999e74

   1. Initial program 91.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.9

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

   7. Applied rewrites 93.9%

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

   if -1.9999999999999999e74 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 9.99999999999999953e67

   1. Initial program 99.5%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.8

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

   8. Applied rewrites 92.8%

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

   if 9.99999999999999953e67 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 85.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 83.4

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

   8. Applied rewrites 83.4%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right) - x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 68000000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y + 0.0007936500793651\right), z \cdot \frac{1}{x}, \log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 68000000000000.0)
   (fma
    (fma
     z
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     0.083333333333333)
    (/ 1.0 x)
    (fma (+ x -0.5) (log x) (- 0.91893853320467 x)))
   (fma
    (* z (+ y 0.0007936500793651))
    (* z (/ 1.0 x))
    (- (* (log x) (+ x -0.5)) (+ x -0.91893853320467)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 68000000000000.0) {
		tmp = fma(fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), (1.0 / x), fma((x + -0.5), log(x), (0.91893853320467 - x)));
	} else {
		tmp = fma((z * (y + 0.0007936500793651)), (z * (1.0 / x)), ((log(x) * (x + -0.5)) - (x + -0.91893853320467)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 68000000000000.0)
		tmp = fma(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), Float64(1.0 / x), fma(Float64(x + -0.5), log(x), Float64(0.91893853320467 - x)));
	else
		tmp = fma(Float64(z * Float64(y + 0.0007936500793651)), Float64(z * Float64(1.0 / x)), Float64(Float64(log(x) * Float64(x + -0.5)) - Float64(x + -0.91893853320467)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 68000000000000.0], N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] * N[(z * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision] - N[(x + -0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.8e13

   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

   4. Applied rewrites 99.8%

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

   if 6.8e13 < 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 z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \left(0.0007936500793651 + y\right), \frac{1}{x} \cdot z, \log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 68000000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y + 0.0007936500793651\right), z \cdot \frac{1}{x}, \log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 70000000000000:\\ \;\;\;\;0.91893853320467 + \left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} + \mathsf{fma}\left(x + -0.5, \log x, -x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y + 0.0007936500793651\right), z \cdot \frac{1}{x}, \log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 70000000000000.0)
   (+
    0.91893853320467
    (+
     (/
      (fma
       z
       (fma (+ y 0.0007936500793651) z -0.0027777777777778)
       0.083333333333333)
      x)
     (fma (+ x -0.5) (log x) (- x))))
   (fma
    (* z (+ y 0.0007936500793651))
    (* z (/ 1.0 x))
    (- (* (log x) (+ x -0.5)) (+ x -0.91893853320467)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 70000000000000.0) {
		tmp = 0.91893853320467 + ((fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x) + fma((x + -0.5), log(x), -x));
	} else {
		tmp = fma((z * (y + 0.0007936500793651)), (z * (1.0 / x)), ((log(x) * (x + -0.5)) - (x + -0.91893853320467)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 70000000000000.0)
		tmp = Float64(0.91893853320467 + Float64(Float64(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x) + fma(Float64(x + -0.5), log(x), Float64(-x))));
	else
		tmp = fma(Float64(z * Float64(y + 0.0007936500793651)), Float64(z * Float64(1.0 / x)), Float64(Float64(log(x) * Float64(x + -0.5)) - Float64(x + -0.91893853320467)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 70000000000000.0], N[(0.91893853320467 + N[(N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] * N[(z * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision] - N[(x + -0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7e13

   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

   4. Applied rewrites 99.8%

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

   if 7e13 < 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 z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \left(0.0007936500793651 + y\right), \frac{1}{x} \cdot z, \log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 70000000000000:\\ \;\;\;\;0.91893853320467 + \left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} + \mathsf{fma}\left(x + -0.5, \log x, -x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y + 0.0007936500793651\right), z \cdot \frac{1}{x}, \log x \cdot \left(x + -0.5\right) - \left(x + -0.91893853320467\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 40000000000:\\ \;\;\;\;0.91893853320467 + \left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} + \mathsf{fma}\left(x + -0.5, \log x, -x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 40000000000.0)
   (+
    0.91893853320467
    (+
     (/
      (fma
       z
       (fma (+ y 0.0007936500793651) z -0.0027777777777778)
       0.083333333333333)
      x)
     (fma (+ x -0.5) (log x) (- x))))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (/ (* z (+ y 0.0007936500793651)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 40000000000.0) {
		tmp = 0.91893853320467 + ((fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x) + fma((x + -0.5), log(x), -x));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * ((z * (y + 0.0007936500793651)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 40000000000.0)
		tmp = Float64(0.91893853320467 + Float64(Float64(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x) + fma(Float64(x + -0.5), log(x), Float64(-x))));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 40000000000.0], N[(0.91893853320467 + N[(N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4e10

   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

   4. Applied rewrites 99.8%

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

   if 4e10 < 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 z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 40000000000:\\ \;\;\;\;0.91893853320467 + \left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} + \mathsf{fma}\left(x + -0.5, \log x, -x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.3)
   (fma
    (fma
     z
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     0.083333333333333)
    (/ 1.0 x)
    (fma -0.5 (log x) 0.91893853320467))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (/ (* z (+ y 0.0007936500793651)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.3) {
		tmp = fma(fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), (1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * ((z * (y + 0.0007936500793651)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.3)
		tmp = fma(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), Float64(1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.3], N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. lower-log.f64 98.1

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

   7. Applied rewrites 98.1%

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

   if 2.2999999999999998 < x

   1. Initial program 85.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+40}:\\ \;\;\;\;0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.8e+40)
   (+
    0.91893853320467
    (/
     (fma
      z
      (fma z (+ y 0.0007936500793651) -0.0027777777777778)
      0.083333333333333)
     x))
   (fma x (log x) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.8e+40) {
		tmp = 0.91893853320467 + (fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	} else {
		tmp = fma(x, log(x), -x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.8e+40)
		tmp = Float64(0.91893853320467 + Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x));
	else
		tmp = fma(x, log(x), Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.8e+40], N[(0.91893853320467 + N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8000000000000001e40

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 93.3

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

   8. Applied rewrites 93.3%

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

   if 2.8000000000000001e40 < x

   1. Initial program 84.5%

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

   3. Taylor expanded in 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. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\log x \cdot x + -1 \cdot x} \]

      7. neg-mul-1 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log x, \mathsf{neg}\left(x\right)\right)} \]

      10. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\log x}, \mathsf{neg}\left(x\right)\right) \]

      11. lower-neg.f64 72.6

          \[\leadsto \mathsf{fma}\left(x, \log x, \color{blue}{-x}\right) \]

   5. Applied rewrites 72.6%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+40}:\\ \;\;\;\;0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+40}:\\ \;\;\;\;0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.8e+40)
   (+
    0.91893853320467
    (/
     (fma
      z
      (fma z (+ y 0.0007936500793651) -0.0027777777777778)
      0.083333333333333)
     x))
   (- (* x (log x)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.8e+40) {
		tmp = 0.91893853320467 + (fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	} else {
		tmp = (x * log(x)) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.8e+40)
		tmp = Float64(0.91893853320467 + Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x));
	else
		tmp = Float64(Float64(x * log(x)) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.8e+40], N[(0.91893853320467 + N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8000000000000001e40

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 93.3

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

   8. Applied rewrites 93.3%

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

   if 2.8000000000000001e40 < x

   1. Initial program 84.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\log x \cdot x + -1 \cdot x} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \log x} + -1 \cdot x \]

      8. neg-mul-1 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 72.4

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

   10. Applied rewrites 72.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+40}:\\ \;\;\;\;0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := z \cdot \left(t\_0 - 0.0027777777777778\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-0.0069444444444443885}{x \cdot \mathsf{fma}\left(z, -0.0027777777777778, -0.083333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651)))
        (t_1 (* z (- t_0 0.0027777777777778))))
   (if (<= t_1 -2e+74)
     (* (* z y) (/ z x))
     (if (<= t_1 5e-6)
       (/
        -0.0069444444444443885
        (* x (fma z -0.0027777777777778 -0.083333333333333)))
       (* z (/ t_0 x))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double t_1 = z * (t_0 - 0.0027777777777778);
	double tmp;
	if (t_1 <= -2e+74) {
		tmp = (z * y) * (z / x);
	} else if (t_1 <= 5e-6) {
		tmp = -0.0069444444444443885 / (x * fma(z, -0.0027777777777778, -0.083333333333333));
	} else {
		tmp = z * (t_0 / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	t_1 = Float64(z * Float64(t_0 - 0.0027777777777778))
	tmp = 0.0
	if (t_1 <= -2e+74)
		tmp = Float64(Float64(z * y) * Float64(z / x));
	elseif (t_1 <= 5e-6)
		tmp = Float64(-0.0069444444444443885 / Float64(x * fma(z, -0.0027777777777778, -0.083333333333333)));
	else
		tmp = Float64(z * Float64(t_0 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(t$95$0 - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+74], N[(N[(z * y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-6], N[(-0.0069444444444443885 / N[(x * N[(z * -0.0027777777777778 + -0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(t$95$0 / x), $MachinePrecision]), $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) < -1.9999999999999999e74

   1. Initial program 91.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.9

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

   7. Applied rewrites 93.9%

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

   if -1.9999999999999999e74 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5.00000000000000041e-6

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. div-inv N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x}}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 98.9

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\frac{-0.0069444444444443885}{x}}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)} \]

   7. Applied rewrites 98.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\frac{-0.0069444444444443885}{x}}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}\right)}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{\color{blue}{x \cdot \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, \mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}} \]

      5. sub-neg N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}, \mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{7936500793651}{10000000000000000} + y}, \frac{-13888888888889}{5000000000000000}\right), \mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)} \]

      9. metadata-eval 44.1

         \[\leadsto \frac{-0.0069444444444443885}{x \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), \color{blue}{-0.083333333333333}\right)} \]

   10. Applied rewrites 44.1%

       \[\leadsto \color{blue}{\frac{-0.0069444444444443885}{x \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \color{blue}{\left(\frac{-13888888888889}{5000000000000000} \cdot z - \frac{83333333333333}{1000000000000000}\right)}} \]
   12. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \color{blue}{\left(\frac{-13888888888889}{5000000000000000} \cdot z + \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \left(\color{blue}{z \cdot \frac{-13888888888889}{5000000000000000}} + \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)\right)} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \left(z \cdot \frac{-13888888888889}{5000000000000000} + \color{blue}{\frac{-83333333333333}{1000000000000000}}\right)} \]

       4. lower-fma.f64 44.1

          \[\leadsto \frac{-0.0069444444444443885}{x \cdot \color{blue}{\mathsf{fma}\left(z, -0.0027777777777778, -0.083333333333333\right)}} \]

   13. Applied rewrites 44.1%

       \[\leadsto \frac{-0.0069444444444443885}{x \cdot \color{blue}{\mathsf{fma}\left(z, -0.0027777777777778, -0.083333333333333\right)}} \]

   if 5.00000000000000041e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   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 z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 79.7

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

   8. Applied rewrites 79.7%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-0.0069444444444443885}{x \cdot \mathsf{fma}\left(z, -0.0027777777777778, -0.083333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-0.0069444444444443885}{x \cdot \mathsf{fma}\left(z, -0.0027777777777778, -0.083333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))))
   (if (<= t_0 -2e+74)
     (* (* z y) (/ z x))
     (if (<= t_0 5e-6)
       (/
        -0.0069444444444443885
        (* x (fma z -0.0027777777777778 -0.083333333333333)))
       (* y (/ (* z z) x))))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -2e+74) {
		tmp = (z * y) * (z / x);
	} else if (t_0 <= 5e-6) {
		tmp = -0.0069444444444443885 / (x * fma(z, -0.0027777777777778, -0.083333333333333));
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))
	tmp = 0.0
	if (t_0 <= -2e+74)
		tmp = Float64(Float64(z * y) * Float64(z / x));
	elseif (t_0 <= 5e-6)
		tmp = Float64(-0.0069444444444443885 / Float64(x * fma(z, -0.0027777777777778, -0.083333333333333)));
	else
		tmp = Float64(y * Float64(Float64(z * z) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+74], N[(N[(z * y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-6], N[(-0.0069444444444443885 / N[(x * N[(z * -0.0027777777777778 + -0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $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) < -1.9999999999999999e74

   1. Initial program 91.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.9

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

   7. Applied rewrites 93.9%

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

   if -1.9999999999999999e74 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5.00000000000000041e-6

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. div-inv N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x}}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 98.9

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\frac{-0.0069444444444443885}{x}}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)} \]

   7. Applied rewrites 98.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\frac{-0.0069444444444443885}{x}}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}\right)}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{\color{blue}{x \cdot \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, \mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)}} \]

      5. sub-neg N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}, \mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{7936500793651}{10000000000000000} + y}, \frac{-13888888888889}{5000000000000000}\right), \mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)} \]

      9. metadata-eval 44.1

         \[\leadsto \frac{-0.0069444444444443885}{x \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), \color{blue}{-0.083333333333333}\right)} \]

   10. Applied rewrites 44.1%

       \[\leadsto \color{blue}{\frac{-0.0069444444444443885}{x \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \color{blue}{\left(\frac{-13888888888889}{5000000000000000} \cdot z - \frac{83333333333333}{1000000000000000}\right)}} \]
   12. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \color{blue}{\left(\frac{-13888888888889}{5000000000000000} \cdot z + \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \left(\color{blue}{z \cdot \frac{-13888888888889}{5000000000000000}} + \left(\mathsf{neg}\left(\frac{83333333333333}{1000000000000000}\right)\right)\right)} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{x \cdot \left(z \cdot \frac{-13888888888889}{5000000000000000} + \color{blue}{\frac{-83333333333333}{1000000000000000}}\right)} \]

       4. lower-fma.f64 44.1

          \[\leadsto \frac{-0.0069444444444443885}{x \cdot \color{blue}{\mathsf{fma}\left(z, -0.0027777777777778, -0.083333333333333\right)}} \]

   13. Applied rewrites 44.1%

       \[\leadsto \frac{-0.0069444444444443885}{x \cdot \color{blue}{\mathsf{fma}\left(z, -0.0027777777777778, -0.083333333333333\right)}} \]

   if 5.00000000000000041e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-0.0069444444444443885}{x \cdot \mathsf{fma}\left(z, -0.0027777777777778, -0.083333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))))
   (if (<= t_0 -2e+74)
     (* (* z y) (/ z x))
     (if (<= t_0 5e-6) (* 0.083333333333333 (/ 1.0 x)) (* y (/ (* z z) x))))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -2e+74) {
		tmp = (z * y) * (z / x);
	} else if (t_0 <= 5e-6) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = y * ((z * z) / 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) :: t_0
    real(8) :: tmp
    t_0 = z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0)
    if (t_0 <= (-2d+74)) then
        tmp = (z * y) * (z / x)
    else if (t_0 <= 5d-6) then
        tmp = 0.083333333333333d0 * (1.0d0 / x)
    else
        tmp = y * ((z * z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double tmp;
	if (t_0 <= -2e+74) {
		tmp = (z * y) * (z / x);
	} else if (t_0 <= 5e-6) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)
	tmp = 0
	if t_0 <= -2e+74:
		tmp = (z * y) * (z / x)
	elif t_0 <= 5e-6:
		tmp = 0.083333333333333 * (1.0 / x)
	else:
		tmp = y * ((z * z) / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))
	tmp = 0.0
	if (t_0 <= -2e+74)
		tmp = Float64(Float64(z * y) * Float64(z / x));
	elseif (t_0 <= 5e-6)
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	else
		tmp = Float64(y * Float64(Float64(z * z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	tmp = 0.0;
	if (t_0 <= -2e+74)
		tmp = (z * y) * (z / x);
	elseif (t_0 <= 5e-6)
		tmp = 0.083333333333333 * (1.0 / x);
	else
		tmp = y * ((z * z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+74], N[(N[(z * y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-6], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $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) < -1.9999999999999999e74

   1. Initial program 91.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.9

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

   7. Applied rewrites 93.9%

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

   if -1.9999999999999999e74 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5.00000000000000041e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 43.9

         \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

   8. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{83333333333333}{1000000000000000}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{83333333333333}{1000000000000000}} \]

      4. lower-/.f64 44.0

         \[\leadsto \color{blue}{\frac{1}{x}} \cdot 0.083333333333333 \]

   10. Applied rewrites 44.0%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   if 5.00000000000000041e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ t_1 := y \cdot \frac{z \cdot z}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
        (t_1 (* y (/ (* z z) x))))
   (if (<= t_0 -2e+74)
     t_1
     (if (<= t_0 5e-6) (* 0.083333333333333 (/ 1.0 x)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double t_1 = y * ((z * z) / x);
	double tmp;
	if (t_0 <= -2e+74) {
		tmp = t_1;
	} else if (t_0 <= 5e-6) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0)
    t_1 = y * ((z * z) / x)
    if (t_0 <= (-2d+74)) then
        tmp = t_1
    else if (t_0 <= 5d-6) then
        tmp = 0.083333333333333d0 * (1.0d0 / x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double t_1 = y * ((z * z) / x);
	double tmp;
	if (t_0 <= -2e+74) {
		tmp = t_1;
	} else if (t_0 <= 5e-6) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)
	t_1 = y * ((z * z) / x)
	tmp = 0
	if t_0 <= -2e+74:
		tmp = t_1
	elif t_0 <= 5e-6:
		tmp = 0.083333333333333 * (1.0 / x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))
	t_1 = Float64(y * Float64(Float64(z * z) / x))
	tmp = 0.0
	if (t_0 <= -2e+74)
		tmp = t_1;
	elseif (t_0 <= 5e-6)
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	t_1 = y * ((z * z) / x);
	tmp = 0.0;
	if (t_0 <= -2e+74)
		tmp = t_1;
	elseif (t_0 <= 5e-6)
		tmp = 0.083333333333333 * (1.0 / x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+74], t$95$1, If[LessEqual[t$95$0, 5e-6], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $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) < -1.9999999999999999e74 or 5.00000000000000041e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 87.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.1

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

   5. Applied rewrites 60.1%

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

   if -1.9999999999999999e74 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5.00000000000000041e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 43.9

         \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

   8. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{83333333333333}{1000000000000000}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{83333333333333}{1000000000000000}} \]

      4. lower-/.f64 44.0

         \[\leadsto \color{blue}{\frac{1}{x}} \cdot 0.083333333333333 \]

   10. Applied rewrites 44.0%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+31}:\\ \;\;\;\;0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7.5e+31)
   (+
    0.91893853320467
    (/
     (fma
      z
      (fma z (+ y 0.0007936500793651) -0.0027777777777778)
      0.083333333333333)
     x))
   (* z (/ (* z (+ y 0.0007936500793651)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7.5e+31) {
		tmp = 0.91893853320467 + (fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	} else {
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7.5e+31)
		tmp = Float64(0.91893853320467 + Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x));
	else
		tmp = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7.5e+31], N[(0.91893853320467 + N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5e31

   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 y around 0

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 93.7

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

   8. Applied rewrites 93.7%

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

   if 7.5e31 < x

   1. Initial program 84.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 32.9

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

   8. Applied rewrites 32.9%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+31}:\\ \;\;\;\;0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7.5e+31)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (* z (/ (* z (+ y 0.0007936500793651)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7.5e+31) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7.5e+31)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7.5e+31], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5e31

   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. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 93.6

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

   5. Applied rewrites 93.6%

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

   if 7.5e31 < x

   1. Initial program 84.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 32.9

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

   8. Applied rewrites 32.9%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 24.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ 0.083333333333333 \cdot \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.083333333333333 (/ 1.0 x)))

C

double code(double x, double y, double z) {
	return 0.083333333333333 * (1.0 / 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 * (1.0d0 / x)
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 * (1.0 / x);
}

Python

def code(x, y, z):
	return 0.083333333333333 * (1.0 / x)

Julia

function code(x, y, z)
	return Float64(0.083333333333333 * Float64(1.0 / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 * (1.0 / x);
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.8%

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

3. Taylor expanded in 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. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. lower--.f64 54.5

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

5. Applied rewrites 54.5%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 21.0

      \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

8. Applied rewrites 21.0%

   \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

   2. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{83333333333333}{1000000000000000}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{83333333333333}{1000000000000000}} \]

   4. lower-/.f64 21.0

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot 0.083333333333333 \]

10. Applied rewrites 21.0%

    \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

11. Final simplification 21.0%

    \[\leadsto 0.083333333333333 \cdot \frac{1}{x} \]
12. Add Preprocessing

Alternative 20: 24.0% 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 92.8%

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

3. Taylor expanded in 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. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. lower--.f64 54.5

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

5. Applied rewrites 54.5%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 21.0

      \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

8. Applied rewrites 21.0%

   \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
9. 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 2024219 
(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)))