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

Average Error: 15.02% → 0.19%

Time: 14.6s

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	15.02%
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 15.02

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

2. Simplified 0.19

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

   [Start] 15.02	\[ \left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   +-commutative [=>] 15.02	\[ \color{blue}{\left(z \cdot \log \left(1 - y\right) + x \cdot \log y\right)} - t \]
   associate--l+ [=>] 15.02	\[ \color{blue}{z \cdot \log \left(1 - y\right) + \left(x \cdot \log y - t\right)} \]
   fma-def [=>] 15.02	\[ \color{blue}{\mathsf{fma}\left(z, \log \left(1 - y\right), x \cdot \log y - t\right)} \]
   sub-neg [=>] 15.02	\[ \mathsf{fma}\left(z, \log \color{blue}{\left(1 + \left(-y\right)\right)}, x \cdot \log y - t\right) \]
   log1p-def [=>] 0.19	\[ \mathsf{fma}\left(z, \color{blue}{\mathsf{log1p}\left(-y\right)}, x \cdot \log y - t\right) \]

3. Final simplification 0.19

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

Alternatives

Alternative 1
Error	0.9%
Cost	13312

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

Alternative 2
Error	9.84%
Cost	7113

\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -6 \cdot 10^{-67} \lor \neg \left(t \leq 2.2 \cdot 10^{-169}\right):\\ \;\;\;\;t_1 - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - z \cdot y\\ \end{array} \]

Alternative 3
Error	10.11%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-87} \lor \neg \left(x \leq 1.2 \cdot 10^{-140}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]

Alternative 4
Error	0.9%
Cost	6976

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

Alternative 5
Error	23.96%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-50} \lor \neg \left(x \leq 6.2 \cdot 10^{+67}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]

Alternative 6
Error	23.88%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-49} \lor \neg \left(x \leq 4.8 \cdot 10^{+65}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \]

Alternative 7
Error	51.17%
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-82}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-165}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 8
Error	42.54%
Cost	384

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

Alternative 9
Error	57.34%
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023121 
(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))