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

Average Error: 9.6 → 0.1

Time: 21.6s

Precision: binary64

Cost: 26048

Math

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

↓

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

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

Target

Original	9.6
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 9.6

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

2. Simplified 0.1

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

Alternatives

Alternative 1
Error	6.5
Cost	6984

\[\begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-0.5 \cdot y\right) + \left(-z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.6
Cost	6976

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

Alternative 3
Error	26.8
Cost	768

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

Alternative 4
Error	26.8
Cost	768

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

Alternative 5
Error	32.7
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-31}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-34}:\\ \;\;\;\;-y \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 6
Error	27.0
Cost	384

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

Alternative 7
Error	36.5
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023010 
(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))