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

Average Error: 3.8 → 1.2

Time: 7.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{a + t}\\ t_2 := \frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\\ \mathbf{if}\;a \leq 6.283594212548852 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(z, \frac{t_1}{t}, \left(b - c\right) \cdot t_2\right)\right)}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(b - c, t_2, t_1 \cdot \frac{z}{t}\right)\right)}, x\right)}\\ \end{array} \]

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
 (let* ((t_1 (sqrt (+ a t)))
        (t_2 (+ (/ 0.6666666666666666 t) (- -0.8333333333333334 a))))
   (if (<= a 6.283594212548852e+25)
     (/ x (fma y (pow (exp 2.0) (fma z (/ t_1 t) (* (- b c) t_2))) x))
     (/ x (fma y (pow (exp 2.0) (fma (- b c) t_2 (* t_1 (/ 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) {
	double t_1 = sqrt((a + t));
	double t_2 = (0.6666666666666666 / t) + (-0.8333333333333334 - a);
	double tmp;
	if (a <= 6.283594212548852e+25) {
		tmp = x / fma(y, pow(exp(2.0), fma(z, (t_1 / t), ((b - c) * t_2))), x);
	} else {
		tmp = x / fma(y, pow(exp(2.0), fma((b - c), t_2, (t_1 * (z / t)))), x);
	}
	return tmp;
}

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)
	t_1 = sqrt(Float64(a + t))
	t_2 = Float64(Float64(0.6666666666666666 / t) + Float64(-0.8333333333333334 - a))
	tmp = 0.0
	if (a <= 6.283594212548852e+25)
		tmp = Float64(x / fma(y, (exp(2.0) ^ fma(z, Float64(t_1 / t), Float64(Float64(b - c) * t_2))), x));
	else
		tmp = Float64(x / fma(y, (exp(2.0) ^ fma(Float64(b - c), t_2, Float64(t_1 * Float64(z / t)))), x));
	end
	return tmp
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_] := Block[{t$95$1 = N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.6666666666666666 / t), $MachinePrecision] + N[(-0.8333333333333334 - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 6.283594212548852e+25], N[(x / N[(y * N[Power[N[Exp[2.0], $MachinePrecision], N[(z * N[(t$95$1 / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[Power[N[Exp[2.0], $MachinePrecision], N[(N[(b - c), $MachinePrecision] * t$95$2 + N[(t$95$1 * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.8
Target	3.0
Herbie	1.2

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

Derivation

1. Split input into 2 regimes

2. if a < 6.2835942125488516e25

   1. Initial program 2.6

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

   2. Simplified 0.6

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

   if 6.2835942125488516e25 < a

   1. Initial program 5.7

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

   2. Simplified 4.9

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

   3. Taylor expanded in z around 0 4.5

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

   4. Simplified 2.1

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.283594212548852 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(z, \frac{\sqrt{a + t}}{t}, \left(b - c\right) \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)\right)}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(b - c, \frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right), \sqrt{a + t} \cdot \frac{z}{t}\right)\right)}, x\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(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)))))))))))