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

Average Error: 9.5 → 0.1

Time: 10.7s

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	9.5
Target	0.3
Herbie	0.1

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

Derivation

1. Initial program 9.5

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

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), x \cdot \log y - t\right)} \]
   Proof
   (fma.f64 z (log1p.f64 (neg.f64 y)) (-.f64 (*.f64 x (log.f64 y)) t)): 0 points increase in error, 0 points decrease in error
   (fma.f64 z (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (neg.f64 y)))) (-.f64 (*.f64 x (log.f64 y)) t)): 51 points increase in error, 0 points decrease in error
   (fma.f64 z (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 y))) (-.f64 (*.f64 x (log.f64 y)) t)): 0 points increase in error, 0 points decrease in error
   (fma.f64 z (log.f64 (-.f64 1 y)) (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 x (log.f64 y)) (neg.f64 t)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (log.f64 (-.f64 1 y))) (+.f64 (*.f64 x (log.f64 y)) (neg.f64 t)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 z (log.f64 (-.f64 1 y))) (*.f64 x (log.f64 y))) (neg.f64 t))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 1 y))))) (neg.f64 t)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 1 y)))) t)): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.3
Cost	7360

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

Alternative 2
Error	6.6
Cost	6984

\[\begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -2.690339324526891 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.9915122738269939 \cdot 10^{-144}:\\ \;\;\;\;z \cdot \left(\left(y \cdot y\right) \cdot \left(-0.5 + y \cdot -0.3333333333333333\right) - y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.5
Cost	6976

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

Alternative 4
Error	26.7
Cost	960

\[z \cdot \left(\left(y \cdot y\right) \cdot \left(-0.5 + y \cdot -0.3333333333333333\right) - y\right) - t \]

Alternative 5
Error	26.8
Cost	704

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

Alternative 6
Error	32.8
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq -2.324160444056339 \cdot 10^{-21}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.4163460429369672 \cdot 10^{-42}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 7
Error	26.9
Cost	384

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

Alternative 8
Error	36.3
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2022317 
(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))