Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I

Average Error: 5.86% → 2.6%

Time: 50.0s

Precision: binary64

Math

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]

↓

\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(-1 \cdot \mathsf{fma}\left(c - b, \frac{0.6666666666666666}{t} - \left(a - -0.8333333333333334\right), \left(-\sqrt{\mathsf{fma}\left(1, a, t\right)}\right) \cdot \frac{z}{t}\right)\right)}, x\right)} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/
  x
  (+
   x
   (*
    y
    (exp
     (*
      2.0
      (-
       (/ (* z (sqrt (+ t a))) t)
       (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))

↓

(FPCore (x y z t a b c)
 :precision binary64
 (/
  x
  (fma
   y
   (pow
    (exp 2.0)
    (*
     -1.0
     (fma
      (- c b)
      (- (/ 0.6666666666666666 t) (- a -0.8333333333333334))
      (* (- (sqrt (fma 1.0 a t))) (/ z t)))))
   x)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	return x / fma(y, pow(exp(2.0), (-1.0 * fma((c - b), ((0.6666666666666666 / t) - (a - -0.8333333333333334)), (-sqrt(fma(1.0, a, t)) * (z / t))))), x);
}

Julia

function code(x, y, z, t, a, b, c)
	return Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))))))))
end

↓

function code(x, y, z, t, a, b, c)
	return Float64(x / fma(y, (exp(2.0) ^ Float64(-1.0 * fma(Float64(c - b), Float64(Float64(0.6666666666666666 / t) - Float64(a - -0.8333333333333334)), Float64(Float64(-sqrt(fma(1.0, a, t))) * Float64(z / t))))), x))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_] := N[(x / N[(y * N[Power[N[Exp[2.0], $MachinePrecision], N[(-1.0 * N[(N[(c - b), $MachinePrecision] * N[(N[(0.6666666666666666 / t), $MachinePrecision] - N[(a - -0.8333333333333334), $MachinePrecision]), $MachinePrecision] + N[((-N[Sqrt[N[(1.0 * a + t), $MachinePrecision]], $MachinePrecision]) * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Target

Original	5.86%
Target	4.5%
Herbie	2.6%

\[\begin{array}{l} \mathbf{if}\;t < -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t < 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array} \]

Derivation

1. Initial program 5.86

   \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]

2. Simplified 5.86

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\frac{\sqrt{t + a} \cdot z}{t} - \left(c - b\right) \cdot \left(\left(\frac{\frac{2}{3}}{t} - a\right) - 0.8333333333333334\right)\right)}, x\right)}} \]
   Proof

3. Applied egg-rr 3.86

   \[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\color{blue}{\left(-1 \cdot \mathsf{fma}\left(c - b, \left(\frac{0.6666666666666666}{t} - a\right) - 0.8333333333333334, -\frac{\sqrt{t + a} \cdot z}{t}\right)\right)}}, x\right)} \]

4. Applied egg-rr 4.62

   \[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(-1 \cdot \color{blue}{\left(\left(c - b\right) \cdot \left(\frac{0.6666666666666666}{t} - \left(a - -0.8333333333333334\right)\right) - \sqrt{\mathsf{fma}\left(1, a, t\right)} \cdot \left(z \cdot \frac{1}{t}\right)\right)}\right)}, x\right)} \]

5. Simplified 2.6

   \[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(-1 \cdot \color{blue}{\mathsf{fma}\left(c - b, \frac{0.6666666666666666}{t} - \left(a - -0.8333333333333334\right), \left(-\sqrt{\mathsf{fma}\left(1, a, t\right)}\right) \cdot \frac{z}{t}\right)}\right)}, x\right)} \]
   Proof

Reproduce

herbie shell --seed 2023136 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2.0 (/ (- (* (* z (sqrt (+ t a))) (* (* 3.0 t) (- a (/ 5.0 6.0)))) (* (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0) (* (- a (/ 5.0 6.0)) (* (- b c) t)))) (* (* (* t t) 3.0) (- a (/ 5.0 6.0))))))))) (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))

  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))