Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Average Error: 15.3 → 0.2

Time: 6.9s

Precision: binary64

Cost: 26112

Math

\[x \cdot \log \left(\frac{x}{y}\right) - z \]

↓

\[\mathsf{fma}\left(x \cdot 3, \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right), -z\right) \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (log (/ x y))) z))

↓

(FPCore (x y z)
 :precision binary64
 (fma (* x 3.0) (log (/ (cbrt x) (cbrt y))) (- z)))

C

double code(double x, double y, double z) {
	return (x * log((x / y))) - z;
}

↓

double code(double x, double y, double z) {
	return fma((x * 3.0), log((cbrt(x) / cbrt(y))), -z);
}

Julia

function code(x, y, z)
	return Float64(Float64(x * log(Float64(x / y))) - z)
end

↓

function code(x, y, z)
	return fma(Float64(x * 3.0), log(Float64(cbrt(x) / cbrt(y))), Float64(-z))
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(x * 3.0), $MachinePrecision] * N[Log[N[(N[Power[x, 1/3], $MachinePrecision] / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-z)), $MachinePrecision]

Target

Original	15.3
Target	7.7
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y < 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Derivation

1. Initial program 15.3

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]

2. Applied egg-rr 15.4

   \[\leadsto x \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{x}{y}}\right)\right)} - z \]

3. Simplified 15.4

   \[\leadsto x \cdot \color{blue}{\left(\log \left(\sqrt[3]{\frac{x}{y}}\right) \cdot 3\right)} - z \]
   Proof
   (*.f64 (log.f64 (cbrt.f64 (/.f64 x y))) 3): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 3 (log.f64 (cbrt.f64 (/.f64 x y))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= metadata-eval (+.f64 2 1)) (log.f64 (cbrt.f64 (/.f64 x y)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 2 (log.f64 (cbrt.f64 (/.f64 x y)))) (log.f64 (cbrt.f64 (/.f64 x y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= log-pow_binary64 (log.f64 (pow.f64 (cbrt.f64 (/.f64 x y)) 2))) (log.f64 (cbrt.f64 (/.f64 x y)))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 15.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 3, \log \left(\sqrt[3]{\frac{x}{y}}\right), -z\right)} \]

5. Applied egg-rr 0.2

   \[\leadsto \mathsf{fma}\left(x \cdot 3, \log \color{blue}{\left(\sqrt[3]{x} \cdot \frac{1}{\sqrt[3]{y}}\right)}, -z\right) \]

6. Simplified 0.2

   \[\leadsto \mathsf{fma}\left(x \cdot 3, \log \color{blue}{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)}, -z\right) \]
   Proof
   (/.f64 (cbrt.f64 x) (cbrt.f64 y)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (cbrt.f64 x) 1)) (cbrt.f64 y)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (cbrt.f64 x) (/.f64 1 (cbrt.f64 y)))): 36 points increase in error, 38 points decrease in error

7. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(x \cdot 3, \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right), -z\right) \]

Alternatives

Alternative 1
Error	8.7
Cost	20424

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+287}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Error	0.2
Cost	19776

\[x \cdot \left(3 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z \]

Alternative 3
Error	6.1
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-123}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 4
Error	0.3
Cost	13508

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 5
Error	21.4
Cost	7048

\[\begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{-46}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 28000:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	21.6
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-46}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 16.5:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Error	32.8
Cost	128

\[-z \]

Alternative 8
Error	62.5
Cost	64

\[z \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))