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

Average Error: 14.52% → 0.2%

Time: 16.2s

Precision: binary64

Cost: 19712

Math

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

↓

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

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

Target

Original	14.52%
Target	0.47%
Herbie	0.2%

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

Derivation

1. Initial program 14.52

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

2. Simplified 0.2

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

   [Start] 14.52	\[ \left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   cancel-sign-sub [<=] 14.52	\[ \color{blue}{\left(x \cdot \log y - \left(-z\right) \cdot \log \left(1 - y\right)\right)} - t \]
   distribute-lft-neg-in [<=] 14.52	\[ \left(x \cdot \log y - \color{blue}{\left(-z \cdot \log \left(1 - y\right)\right)}\right) - t \]
   fma-neg [=>] 14.52	\[ \color{blue}{\mathsf{fma}\left(x, \log y, -\left(-z \cdot \log \left(1 - y\right)\right)\right)} - t \]
   remove-double-neg [=>] 14.52	\[ \mathsf{fma}\left(x, \log y, \color{blue}{z \cdot \log \left(1 - y\right)}\right) - t \]
   sub-neg [=>] 14.52	\[ \mathsf{fma}\left(x, \log y, z \cdot \log \color{blue}{\left(1 + \left(-y\right)\right)}\right) - t \]
   log1p-def [=>] 0.2	\[ \mathsf{fma}\left(x, \log y, z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)}\right) - t \]

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2%
Cost	13440

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

Alternative 2
Error	0.94%
Cost	7360

\[\left(y \cdot z + \left(x \cdot \log y + \left(y \cdot z\right) \cdot -2\right)\right) - t \]

Alternative 3
Error	0.58%
Cost	7360

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

Alternative 4
Error	11.75%
Cost	7108

\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -3.65 \cdot 10^{-141}:\\ \;\;\;\;\left(t + t_1\right) + t \cdot -2\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-244}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \]

Alternative 5
Error	11.73%
Cost	7049

\[\begin{array}{l} \mathbf{if}\;x \leq -1.36 \cdot 10^{-140} \lor \neg \left(x \leq 1.3 \cdot 10^{-244}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 6
Error	11.93%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-140} \lor \neg \left(x \leq 1.4 \cdot 10^{-244}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]

Alternative 7
Error	0.94%
Cost	6976

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

Alternative 8
Error	22.24%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+29} \lor \neg \left(x \leq 1.45 \cdot 10^{+38}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]

Alternative 9
Error	52.16%
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-130}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-112}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 10
Error	43.8%
Cost	384

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

Alternative 11
Error	58.06%
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023088 
(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))