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

Average Accuracy: 84.8% → 99.8%

Time: 12.5s

Precision: binary64

Cost: 19712

Math

\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]

↓

\[\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), x \cdot \log y - t\right) \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))

↓

(FPCore (x y z t)
 :precision binary64
 (fma z (log1p (- y)) (- (* x (log y)) t)))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * log((1.0 - y)))) - t;
}

↓

double code(double x, double y, double z, double t) {
	return fma(z, log1p(-y), ((x * log(y)) - t));
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(z * log(Float64(1.0 - y)))) - t)
end

↓

function code(x, y, z, t)
	return fma(z, log1p(Float64(-y)), Float64(Float64(x * log(y)) - t))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(z * N[Log[1 + (-y)], $MachinePrecision] + N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

Target

Original	84.8%
Target	99.6%
Herbie	99.8%

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right) \]

Derivation

1. Initial program 84.8%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]

2. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), x \cdot \log y - t\right)} \]
   Proof

   [Start] 84.8	\[ \left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   +-commutative [=>] 84.8	\[ \color{blue}{\left(z \cdot \log \left(1 - y\right) + x \cdot \log y\right)} - t \]
   associate--l+ [=>] 84.8	\[ \color{blue}{z \cdot \log \left(1 - y\right) + \left(x \cdot \log y - t\right)} \]
   fma-def [=>] 84.8	\[ \color{blue}{\mathsf{fma}\left(z, \log \left(1 - y\right), x \cdot \log y - t\right)} \]
   sub-neg [=>] 84.8	\[ \mathsf{fma}\left(z, \log \color{blue}{\left(1 + \left(-y\right)\right)}, x \cdot \log y - t\right) \]
   log1p-def [=>] 99.8	\[ \mathsf{fma}\left(z, \color{blue}{\mathsf{log1p}\left(-y\right)}, x \cdot \log y - t\right) \]

3. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), x \cdot \log y - t\right) \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	19712

\[\mathsf{fma}\left(x, \log y, z \cdot \mathsf{log1p}\left(-y\right)\right) - t \]

Alternative 2
Accuracy	99.2%
Cost	13312

\[\mathsf{fma}\left(\log y, x, y \cdot \left(-z\right)\right) - t \]

Alternative 3
Accuracy	77.5%
Cost	7121

\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{+112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{+60}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \mathbf{elif}\;x \leq -5.9 \cdot 10^{-15} \lor \neg \left(x \leq 8.8 \cdot 10^{+95}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(y \cdot \left(y \cdot -0.5\right)\right) - t\right) - z \cdot y\\ \end{array} \]

Alternative 4
Accuracy	89.9%
Cost	7049

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-88} \lor \neg \left(x \leq 1.7 \cdot 10^{-189}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 5
Accuracy	89.8%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-81} \lor \neg \left(x \leq 1.8 \cdot 10^{-189}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(y \cdot \left(y \cdot -0.5\right)\right) - t\right) - z \cdot y\\ \end{array} \]

Alternative 6
Accuracy	99.2%
Cost	6976

\[\left(x \cdot \log y - z \cdot y\right) - t \]

Alternative 7
Accuracy	58.7%
Cost	832

\[\left(z \cdot \left(y \cdot \left(y \cdot -0.5\right)\right) - t\right) - z \cdot y \]

Alternative 8
Accuracy	49.4%
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-89}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 9
Accuracy	58.4%
Cost	384

\[y \cdot \left(-z\right) - t \]

Alternative 10
Accuracy	43.5%
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023133 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))