Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A

Average Error: 0.1 → 0.1

Time: 8.2s

Precision: binary64

Cost: 19648

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t) :precision binary64 (fma (log y) x (- (- (log t) y) z)))

C

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

↓

double code(double x, double y, double z, double t) {
	return fma(log(y), x, ((log(t) - y) - z));
}

Julia

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

↓

function code(x, y, z, t)
	return fma(log(y), x, Float64(Float64(log(t) - y) - z))
end

Wolfram

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

↓

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

   \[\leadsto \color{blue}{x \cdot \log y - \left(y + \left(z - \log t\right)\right)} \]
   Proof
   (-.f64 (*.f64 x (log.f64 y)) (+.f64 y (-.f64 z (log.f64 t)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--l-_binary64 (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) (-.f64 z (log.f64 t)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) (log.f64 t))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	8.8
Cost	33872

\[\begin{array}{l} t_1 := \log y \cdot x\\ t_2 := t_1 - y\\ t_3 := t_1 - z\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{+123}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	0.1
Cost	13376

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

Alternative 3
Error	0.7
Cost	7112

\[\begin{array}{l} t_1 := \log y \cdot x - \left(y + z\right)\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-8}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	10.5
Cost	6984

\[\begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -1.16 \cdot 10^{+169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	7.0
Cost	6984

\[\begin{array}{l} t_1 := \log y \cdot x - y\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+29}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	18.4
Cost	6856

\[\begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -1.42 \cdot 10^{+169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+85}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	32.9
Cost	260

\[\begin{array}{l} \mathbf{if}\;y \leq 59000000000000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 8
Error	26.3
Cost	256

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

Alternative 9
Error	45.1
Cost	128

\[-y \]

Reproduce

herbie shell --seed 2022330 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))