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

Average Error: 0.13% → 0.13%

Time: 13.3s

Precision: binary64

Cost: 39488

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(x, \log y, y\right)\\ \frac{t_1}{\frac{t_1}{x \cdot \log y - y}} + \left(\log t - z\right) \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma x (log y) y)))
   (+ (/ t_1 (/ t_1 (- (* x (log y)) y))) (- (log t) 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) {
	double t_1 = fma(x, log(y), y);
	return (t_1 / (t_1 / ((x * log(y)) - y))) + (log(t) - 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)
	t_1 = fma(x, log(y), y)
	return Float64(Float64(t_1 / Float64(t_1 / Float64(Float64(x * log(y)) - y))) + Float64(log(t) - 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_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision] + y), $MachinePrecision]}, N[(N[(t$95$1 / N[(t$95$1 / N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.13

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

2. Simplified 0.13

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

   [Start] 0.13	\[ \left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   associate-+l- [=>] 0.13	\[ \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]

3. Applied egg-rr 0.13

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

4. Final simplification 0.13

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

Alternatives

Alternative 1
Error	0.13%
Cost	13376

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

Alternative 2
Error	34.57%
Cost	7516

\[\begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(-z\right) - y\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1160000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-249}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-83}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+167}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	31.54%
Cost	7384

\[\begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(-z\right) - y\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+99}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-272}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-81}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+167}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	16.68%
Cost	7250

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+152} \lor \neg \left(x \leq -2.4 \cdot 10^{+104} \lor \neg \left(x \leq -5.7 \cdot 10^{+76}\right) \land x \leq 1.2 \cdot 10^{+168}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 5
Error	29.46%
Cost	7122

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+151} \lor \neg \left(x \leq -1 \cdot 10^{+98} \lor \neg \left(x \leq -3.2 \cdot 10^{+76}\right) \land x \leq 7.5 \cdot 10^{+167}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 6
Error	0.69%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 7
Error	10.98%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+22} \lor \neg \left(x \leq 5.8 \cdot 10^{+109}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 8
Error	10.7%
Cost	6984

\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.82 \cdot 10^{+22}:\\ \;\;\;\;t_1 - y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+115}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - z\\ \end{array} \]

Alternative 9
Error	51.02%
Cost	260

\[\begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+37}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 10
Error	41.93%
Cost	256

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

Alternative 11
Error	70.48%
Cost	128

\[-y \]

Reproduce

herbie shell --seed 2023088 
(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)))