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

Average Error: 14.04% → 0.19%

Time: 17.9s

Precision: binary64

Cost: 26048

Math

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

↓

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

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 x (log y) (fma z (log1p (- 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(x, log(y), fma(z, log1p(-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(x, log(y), fma(z, log1p(Float64(-y)), Float64(-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[(x * N[Log[y], $MachinePrecision] + N[(z * N[Log[1 + (-y)], $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]

Target

Original	14.04%
Target	0.42%
Herbie	0.19%

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

Derivation

1. Initial program 14.04

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

2. Simplified 0.19

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

   [Start] 14.04	\[ \left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   associate--l+ [=>] 14.04	\[ \color{blue}{x \cdot \log y + \left(z \cdot \log \left(1 - y\right) - t\right)} \]
   fma-def [=>] 14.04	\[ \color{blue}{\mathsf{fma}\left(x, \log y, z \cdot \log \left(1 - y\right) - t\right)} \]
   fma-neg [=>] 14.04	\[ \mathsf{fma}\left(x, \log y, \color{blue}{\mathsf{fma}\left(z, \log \left(1 - y\right), -t\right)}\right) \]
   sub-neg [=>] 14.04	\[ \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(z, \log \color{blue}{\left(1 + \left(-y\right)\right)}, -t\right)\right) \]
   log1p-def [=>] 0.19	\[ \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(z, \color{blue}{\mathsf{log1p}\left(-y\right)}, -t\right)\right) \]

3. Final simplification 0.19

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

Alternatives

Alternative 1
Error	0.2%
Cost	19712

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

Alternative 2
Error	0.87%
Cost	13312

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

Alternative 3
Error	15.8%
Cost	7112

\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{-35}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-113}:\\ \;\;\;\;t_1 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \]

Alternative 4
Error	9.78%
Cost	7049

\[\begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-24} \lor \neg \left(x \leq 7.2 \cdot 10^{-45}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 5
Error	10.21%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-21} \lor \neg \left(x \leq 5.1 \cdot 10^{-48}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \]

Alternative 6
Error	0.87%
Cost	6976

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

Alternative 7
Error	23.24%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -6.9 \cdot 10^{-16} \lor \neg \left(x \leq 6.8 \cdot 10^{+98}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \]

Alternative 8
Error	51.81%
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-43}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-112}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 9
Error	43.48%
Cost	384

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

Alternative 10
Error	57.2%
Cost	128

\[-t \]

Alternative 11
Error	97.73%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023089 
(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))