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

Average Error: 6.0 → 0.6

Time: 16.8s

Precision: binary64

Cost: 14020

Math

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

↓

\[\begin{array}{l} t_0 := \log x \cdot \left(x + -0.5\right)\\ \mathbf{if}\;z \leq -0.0058:\\ \;\;\;\;0.91893853320467 + \left(\mathsf{fma}\left(x + -0.5, \log x, \frac{z}{\frac{x}{z}} \cdot \left(0.0007936500793651 + y\right)\right) - x\right)\\ \mathbf{elif}\;z \leq 2400000:\\ \;\;\;\;\left(0.91893853320467 + \left(t_0 - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(0.0007936500793651 + y\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \left(\left(\frac{0.083333333333333}{x} + \left(t_0 + z \cdot \frac{z}{\frac{x}{0.0007936500793651 + y}}\right)\right) - x\right)\\ \end{array} \]

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)))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log x) (+ x -0.5))))
   (if (<= z -0.0058)
     (+
      0.91893853320467
      (-
       (fma (+ x -0.5) (log x) (* (/ z (/ x z)) (+ 0.0007936500793651 y)))
       x))
     (if (<= z 2400000.0)
       (+
        (+ 0.91893853320467 (- t_0 x))
        (/
         (+
          0.083333333333333
          (* z (+ -0.0027777777777778 (* z (+ 0.0007936500793651 y)))))
         x))
       (+
        0.91893853320467
        (-
         (+
          (/ 0.083333333333333 x)
          (+ t_0 (* z (/ z (/ x (+ 0.0007936500793651 y))))))
         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);
}

↓

double code(double x, double y, double z) {
	double t_0 = log(x) * (x + -0.5);
	double tmp;
	if (z <= -0.0058) {
		tmp = 0.91893853320467 + (fma((x + -0.5), log(x), ((z / (x / z)) * (0.0007936500793651 + y))) - x);
	} else if (z <= 2400000.0) {
		tmp = (0.91893853320467 + (t_0 - x)) + ((0.083333333333333 + (z * (-0.0027777777777778 + (z * (0.0007936500793651 + y))))) / x);
	} else {
		tmp = 0.91893853320467 + (((0.083333333333333 / x) + (t_0 + (z * (z / (x / (0.0007936500793651 + y)))))) - x);
	}
	return tmp;
}

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

↓

function code(x, y, z)
	t_0 = Float64(log(x) * Float64(x + -0.5))
	tmp = 0.0
	if (z <= -0.0058)
		tmp = Float64(0.91893853320467 + Float64(fma(Float64(x + -0.5), log(x), Float64(Float64(z / Float64(x / z)) * Float64(0.0007936500793651 + y))) - x));
	elseif (z <= 2400000.0)
		tmp = Float64(Float64(0.91893853320467 + Float64(t_0 - x)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(-0.0027777777777778 + Float64(z * Float64(0.0007936500793651 + y))))) / x));
	else
		tmp = Float64(0.91893853320467 + Float64(Float64(Float64(0.083333333333333 / x) + Float64(t_0 + Float64(z * Float64(z / Float64(x / Float64(0.0007936500793651 + y)))))) - x));
	end
	return tmp
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]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.0058], N[(0.91893853320467 + N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(z / N[(x / z), $MachinePrecision]), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2400000.0], N[(N[(0.91893853320467 + N[(t$95$0 - x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(-0.0027777777777778 + N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(0.91893853320467 + N[(N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(t$95$0 + N[(z * N[(z / N[(x / N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.0
Target	1.3
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if z < -0.0058

   1. Initial program 21.4

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

   2. Simplified 21.4

      \[\leadsto \color{blue}{0.91893853320467 + \left(\mathsf{fma}\left(x + -0.5, \log x, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right) - x\right)} \]
      Proof
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (+.f64 x -1/2) (log.f64 x) (/.f64 (fma.f64 z (fma.f64 (+.f64 y 7936500793651/10000000000000000) z -13888888888889/5000000000000000) 83333333333333/1000000000000000) x)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1/2))) (log.f64 x) (/.f64 (fma.f64 z (fma.f64 (+.f64 y 7936500793651/10000000000000000) z -13888888888889/5000000000000000) 83333333333333/1000000000000000) x)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1/2)) (log.f64 x) (/.f64 (fma.f64 z (fma.f64 (+.f64 y 7936500793651/10000000000000000) z -13888888888889/5000000000000000) 83333333333333/1000000000000000) x)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (-.f64 x 1/2) (log.f64 x) (/.f64 (fma.f64 z (fma.f64 (+.f64 y 7936500793651/10000000000000000) z (Rewrite<= metadata-eval (neg.f64 13888888888889/5000000000000000))) 83333333333333/1000000000000000) x)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (-.f64 x 1/2) (log.f64 x) (/.f64 (fma.f64 z (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000)) 83333333333333/1000000000000000) x)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (-.f64 x 1/2) (log.f64 x) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000)) 83333333333333/1000000000000000)) x)) x)): 0 points increase in error, 1 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (-.f64 x 1/2) (log.f64 x) (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z)) 83333333333333/1000000000000000) x)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (-.f64 x 1/2) (log.f64 x) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))))) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))) x)): 1 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) (+.f64 (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (Rewrite=> associate--r+_binary64 (-.f64 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) x) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) (neg.f64 x))) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 x) (*.f64 (-.f64 x 1/2) (log.f64 x)))) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 x)) (*.f64 (-.f64 x 1/2) (log.f64 x))) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.f64 x (*.f64 (-.f64 x 1/2) (log.f64 x))))) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (Rewrite=> associate--l-_binary64 (-.f64 0 (+.f64 (-.f64 x (*.f64 (-.f64 x 1/2) (log.f64 x))) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 0 (Rewrite<= sub-neg_binary64 (-.f64 (-.f64 x (*.f64 (-.f64 x 1/2) (log.f64 x))) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (-.f64 x (*.f64 (-.f64 x 1/2) (log.f64 x)))) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (+.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 x) (*.f64 (-.f64 x 1/2) (log.f64 x)))) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (+.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 x)) (*.f64 (-.f64 x 1/2) (log.f64 x))) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) (neg.f64 x))) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) x)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 91893853320467/100000000000000 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) x)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))): 3 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) x) 91893853320467/100000000000000)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in z around inf 22.7

      \[\leadsto 0.91893853320467 + \left(\mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}}\right) - x\right) \]

   4. Simplified 2.1

      \[\leadsto 0.91893853320467 + \left(\mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\frac{z}{\frac{x}{z}} \cdot \left(y + 0.0007936500793651\right)}\right) - x\right) \]
      Proof
      (*.f64 (/.f64 z (/.f64 x z)) (+.f64 y 7936500793651/10000000000000000)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z z) x)) (+.f64 y 7936500793651/10000000000000000)): 52 points increase in error, 19 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) x) (+.f64 y 7936500793651/10000000000000000)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 z 2) x) (Rewrite<= +-commutative_binary64 (+.f64 7936500793651/10000000000000000 y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 z 2) (/.f64 x (+.f64 7936500793651/10000000000000000 y)))): 37 points increase in error, 19 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 z 2) (+.f64 7936500793651/10000000000000000 y)) x)): 34 points increase in error, 31 points decrease in error

   if -0.0058 < z < 2.4e6

   1. Initial program 0.4

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

   if 2.4e6 < z

   1. Initial program 21.5

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

   2. Simplified 21.5

      \[\leadsto \color{blue}{0.91893853320467 + \left(\mathsf{fma}\left(x + -0.5, \log x, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right) - x\right)} \]
      Proof
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (+.f64 x -1/2) (log.f64 x) (/.f64 (fma.f64 z (fma.f64 (+.f64 y 7936500793651/10000000000000000) z -13888888888889/5000000000000000) 83333333333333/1000000000000000) x)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1/2))) (log.f64 x) (/.f64 (fma.f64 z (fma.f64 (+.f64 y 7936500793651/10000000000000000) z -13888888888889/5000000000000000) 83333333333333/1000000000000000) x)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1/2)) (log.f64 x) (/.f64 (fma.f64 z (fma.f64 (+.f64 y 7936500793651/10000000000000000) z -13888888888889/5000000000000000) 83333333333333/1000000000000000) x)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (-.f64 x 1/2) (log.f64 x) (/.f64 (fma.f64 z (fma.f64 (+.f64 y 7936500793651/10000000000000000) z (Rewrite<= metadata-eval (neg.f64 13888888888889/5000000000000000))) 83333333333333/1000000000000000) x)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (-.f64 x 1/2) (log.f64 x) (/.f64 (fma.f64 z (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000)) 83333333333333/1000000000000000) x)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (-.f64 x 1/2) (log.f64 x) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000)) 83333333333333/1000000000000000)) x)) x)): 0 points increase in error, 1 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (-.f64 x 1/2) (log.f64 x) (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z)) 83333333333333/1000000000000000) x)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (fma.f64 (-.f64 x 1/2) (log.f64 x) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))))) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))) x)): 1 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) (+.f64 (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (Rewrite=> associate--r+_binary64 (-.f64 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) x) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) (neg.f64 x))) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 x) (*.f64 (-.f64 x 1/2) (log.f64 x)))) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 x)) (*.f64 (-.f64 x 1/2) (log.f64 x))) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.f64 x (*.f64 (-.f64 x 1/2) (log.f64 x))))) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (Rewrite=> associate--l-_binary64 (-.f64 0 (+.f64 (-.f64 x (*.f64 (-.f64 x 1/2) (log.f64 x))) (neg.f64 (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (-.f64 0 (Rewrite<= sub-neg_binary64 (-.f64 (-.f64 x (*.f64 (-.f64 x 1/2) (log.f64 x))) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (-.f64 x (*.f64 (-.f64 x 1/2) (log.f64 x)))) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (+.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 x) (*.f64 (-.f64 x 1/2) (log.f64 x)))) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (+.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 x)) (*.f64 (-.f64 x 1/2) (log.f64 x))) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) (neg.f64 x))) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 91893853320467/100000000000000 (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) x)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 91893853320467/100000000000000 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) x)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x))): 3 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 (*.f64 (-.f64 x 1/2) (log.f64 x)) x) 91893853320467/100000000000000)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) 83333333333333/1000000000000000) x)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 21.5

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

   4. Simplified 3.9

      \[\leadsto 0.91893853320467 + \left(\color{blue}{\left(\frac{0.083333333333333}{x} + \left(\frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{x} \cdot z + \log x \cdot \left(x - 0.5\right)\right)\right)} - x\right) \]
      Proof
      (+.f64 (/.f64 83333333333333/1000000000000000 x) (+.f64 (*.f64 (/.f64 (fma.f64 z (+.f64 7936500793651/10000000000000000 y) -13888888888889/5000000000000000) x) z) (*.f64 (log.f64 x) (-.f64 x 1/2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 83333333333333/1000000000000000 1)) x) (+.f64 (*.f64 (/.f64 (fma.f64 z (+.f64 7936500793651/10000000000000000 y) -13888888888889/5000000000000000) x) z) (*.f64 (log.f64 x) (-.f64 x 1/2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x))) (+.f64 (*.f64 (/.f64 (fma.f64 z (+.f64 7936500793651/10000000000000000 y) -13888888888889/5000000000000000) x) z) (*.f64 (log.f64 x) (-.f64 x 1/2)))): 14 points increase in error, 7 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (*.f64 (/.f64 (fma.f64 z (+.f64 7936500793651/10000000000000000 y) (Rewrite<= metadata-eval (neg.f64 13888888888889/5000000000000000))) x) z) (*.f64 (log.f64 x) (-.f64 x 1/2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (*.f64 (/.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000)) x) z) (*.f64 (log.f64 x) (-.f64 x 1/2)))): 1 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) (/.f64 x z))) (*.f64 (log.f64 x) (-.f64 x 1/2)))): 8 points increase in error, 12 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x)) (*.f64 (log.f64 x) (-.f64 x 1/2)))): 23 points increase in error, 5 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 (log.f64 x) (Rewrite=> sub-neg_binary64 (+.f64 x (neg.f64 1/2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 (log.f64 x) (+.f64 x (Rewrite=> metadata-eval -1/2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (log.f64 x) x) (*.f64 (log.f64 x) -1/2))))): 1 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (+.f64 (*.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 x)))) x) (*.f64 (log.f64 x) -1/2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (+.f64 (*.f64 (neg.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 x)))) x) (*.f64 (log.f64 x) -1/2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (+.f64 (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 (log.f64 (/.f64 1 x)) x))) (*.f64 (log.f64 x) -1/2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (+.f64 (neg.f64 (*.f64 (log.f64 (/.f64 1 x)) x)) (*.f64 (log.f64 x) (Rewrite<= metadata-eval (neg.f64 1/2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (+.f64 (neg.f64 (*.f64 (log.f64 (/.f64 1 x)) x)) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (log.f64 x) 1/2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x))) (neg.f64 (*.f64 (log.f64 x) 1/2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (+.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x)) (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (log.f64 x)) 1/2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (+.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x)) (*.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 x))) 1/2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (+.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x)) (Rewrite<= *-commutative_binary64 (*.f64 1/2 (log.f64 (/.f64 1 x))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 1/2 (log.f64 (/.f64 1 x))) (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 1/2 (log.f64 (/.f64 1 x)))) (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 83333333333333/1000000000000000 (/.f64 1 x)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 1/2 (log.f64 (/.f64 1 x))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 1/2 (log.f64 (/.f64 1 x))))) (*.f64 83333333333333/1000000000000000 (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 1/2 (log.f64 (/.f64 1 x)))) (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x)))) (*.f64 83333333333333/1000000000000000 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 1/2 (log.f64 (/.f64 1 x))) (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x))) (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x))) (*.f64 83333333333333/1000000000000000 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-+l+_binary64 (+.f64 (*.f64 1/2 (log.f64 (/.f64 1 x))) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x))))) (*.f64 83333333333333/1000000000000000 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 1/2 (Rewrite=> log-rec_binary64 (neg.f64 (log.f64 x)))) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x)))) (*.f64 83333333333333/1000000000000000 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 1/2 (log.f64 x)))) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x)))) (*.f64 83333333333333/1000000000000000 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 1/2) (log.f64 x))) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x)))) (*.f64 83333333333333/1000000000000000 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 (Rewrite=> metadata-eval -1/2) (log.f64 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 -1 (*.f64 (log.f64 (/.f64 1 x)) x)))) (*.f64 83333333333333/1000000000000000 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1/2 (log.f64 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (Rewrite=> mul-1-neg_binary64 (neg.f64 (*.f64 (log.f64 (/.f64 1 x)) x))))) (*.f64 83333333333333/1000000000000000 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1/2 (log.f64 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (log.f64 (/.f64 1 x))) x)))) (*.f64 83333333333333/1000000000000000 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1/2 (log.f64 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 (neg.f64 (Rewrite=> log-rec_binary64 (neg.f64 (log.f64 x)))) x))) (*.f64 83333333333333/1000000000000000 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1/2 (log.f64 x)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 (Rewrite=> remove-double-neg_binary64 (log.f64 x)) x))) (*.f64 83333333333333/1000000000000000 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -1/2 (log.f64 x)) (+.f64 (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (*.f64 (log.f64 x) x)) (*.f64 83333333333333/1000000000000000 (/.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (log.f64 x)) (Rewrite<= associate-+r+_binary64 (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) 13888888888889/5000000000000000) z) x) (+.f64 (*.f64 (log.f64 x) x) (*.f64 83333333333333/1000000000000000 (/.f64 1 x)))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in z around inf 4.2

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

   6. Simplified 0.7

      \[\leadsto 0.91893853320467 + \left(\left(\frac{0.083333333333333}{x} + \left(\color{blue}{\frac{z}{\frac{x}{y + 0.0007936500793651}}} \cdot z + \log x \cdot \left(x - 0.5\right)\right)\right) - x\right) \]
      Proof
      (/.f64 z (/.f64 x (+.f64 y 7936500793651/10000000000000000))): 0 points increase in error, 0 points decrease in error
      (/.f64 z (/.f64 x (Rewrite<= +-commutative_binary64 (+.f64 7936500793651/10000000000000000 y)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z (+.f64 7936500793651/10000000000000000 y)) x)): 38 points increase in error, 56 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0058:\\ \;\;\;\;0.91893853320467 + \left(\mathsf{fma}\left(x + -0.5, \log x, \frac{z}{\frac{x}{z}} \cdot \left(0.0007936500793651 + y\right)\right) - x\right)\\ \mathbf{elif}\;z \leq 2400000:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(0.0007936500793651 + y\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \left(\left(\frac{0.083333333333333}{x} + \left(\log x \cdot \left(x + -0.5\right) + z \cdot \frac{z}{\frac{x}{0.0007936500793651 + y}}\right)\right) - x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.2
Cost	20736

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

Alternative 2
Error	2.2
Cost	14272

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

Alternative 3
Error	0.4
Cost	8004

\[\begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-34}:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(0.0007936500793651 + y\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \left(\left(\frac{0.083333333333333}{x} + \left(\log x \cdot \left(x + -0.5\right) + z \cdot \frac{z}{\frac{x}{0.0007936500793651 + y}}\right)\right) - x\right)\\ \end{array} \]

Alternative 4
Error	1.2
Cost	7748

\[\begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(0.0007936500793651 + y\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \left(\left(\log x \cdot \left(x + -0.5\right) + z \cdot \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right)\right) - x\right)\\ \end{array} \]

Alternative 5
Error	6.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 3700000000:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(0.0007936500793651 + y\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \log x\right)\\ \end{array} \]

Alternative 6
Error	31.6
Cost	1096

\[\begin{array}{l} t_0 := 0.91893853320467 + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \mathbf{if}\;z \leq -10.2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333 + y \cdot \left(z \cdot z\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	31.2
Cost	1092

\[\begin{array}{l} \mathbf{if}\;x \leq 380000:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(0.0007936500793651 + y\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 8
Error	36.3
Cost	968

\[\begin{array}{l} t_0 := 0.91893853320467 + \left(0.0007936500793651 + y\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-34}:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	36.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-18}:\\ \;\;\;\;0.91893853320467 + \frac{z \cdot z}{\frac{x}{0.0007936500793651 + y}}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-34}:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \left(0.0007936500793651 + y\right) \cdot \frac{z \cdot z}{x}\\ \end{array} \]

Alternative 10
Error	34.1
Cost	968

\[\begin{array}{l} t_0 := 0.91893853320467 + \left(0.0007936500793651 + y\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{if}\;z \leq -3 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10.2:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333 + y \cdot \left(z \cdot z\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	31.6
Cost	964

\[\begin{array}{l} \mathbf{if}\;x \leq 2150:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333 + \left(0.0007936500793651 + y\right) \cdot \left(z \cdot z\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 12
Error	39.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq 0.0046:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + y \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]

Alternative 13
Error	40.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq 0.007:\\ \;\;\;\;0.91893853320467 + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + z \cdot \frac{z \cdot y}{x}\\ \end{array} \]

Alternative 14
Error	42.1
Cost	320

\[0.91893853320467 + \frac{0.083333333333333}{x} \]

Alternative 15
Error	60.9
Cost	64

\[0.91893853320467 \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

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

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