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

?

Percentage Accurate: 96.4% → 99.5%
Time: 21.0s
Precision: binary64
Cost: 26368

?

\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
↓
\[x \cdot e^{\mathsf{fma}\left(a, \mathsf{log1p}\left(-z\right) - b, y \cdot \left(\log z - t\right)\right)} \]
(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))
↓
(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (fma a (- (log1p (- z)) b) (* y (- (log z) t))))))
double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(fma(a, (log1p(-z) - b), (y * (log(z) - t))));
}

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end
↓
function code(x, y, z, t, a, b)
	return Float64(x * exp(fma(a, Float64(log1p(Float64(-z)) - b), Float64(y * Float64(log(z) - t)))))
end

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(a * N[(N[Log[1 + (-z)], $MachinePrecision] - b), $MachinePrecision] + N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{\mathsf{fma}\left(a, \mathsf{log1p}\left(-z\right) - b, y \cdot \left(\log z - t\right)\right)}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 20 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Derivation?

  1. Initial program 96.9%

    \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

  2. Simplified99.2%

    \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(a, \mathsf{log1p}\left(-z\right) - b, y \cdot \left(\log z - t\right)\right)}} \]
    Step-by-step derivation

    [Start]96.9%

    \[ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

    +-commutative [=>]96.9%

    \[ x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right) + y \cdot \left(\log z - t\right)}} \]

    fma-def [=>]97.7%

    \[ x \cdot e^{\color{blue}{\mathsf{fma}\left(a, \log \left(1 - z\right) - b, y \cdot \left(\log z - t\right)\right)}} \]

    sub-neg [=>]97.7%

    \[ x \cdot e^{\mathsf{fma}\left(a, \log \color{blue}{\left(1 + \left(-z\right)\right)} - b, y \cdot \left(\log z - t\right)\right)} \]

    log1p-def [=>]99.2%

    \[ x \cdot e^{\mathsf{fma}\left(a, \color{blue}{\mathsf{log1p}\left(-z\right)} - b, y \cdot \left(\log z - t\right)\right)} \]

  3. Final simplification99.2%

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

Alternatives

Alternative 1
Accuracy99.5%
Cost26368
\[x \cdot e^{\mathsf{fma}\left(a, \mathsf{log1p}\left(-z\right) - b, y \cdot \left(\log z - t\right)\right)} \]
Alternative 2
Accuracy96.4%
Cost20160
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
Alternative 3
Accuracy85.7%
Cost13776
\[\begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{if}\;y \leq -7.4 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-134}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-86}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq 0.9:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Accuracy72.3%
Cost7576
\[\begin{array}{l} t_1 := x \cdot e^{a \cdot \left(-b\right)}\\ t_2 := x \cdot e^{t \cdot \left(-y\right)}\\ t_3 := x \cdot {z}^{y}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{+50}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-207}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-49}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 5
Accuracy72.0%
Cost7308
\[\begin{array}{l} t_1 := x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{if}\;a \leq -3.7 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.75 \cdot 10^{-223}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+36}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Accuracy53.9%
Cost7049
\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+60} \lor \neg \left(a \leq 3.1 \cdot 10^{+171}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 7
Accuracy73.8%
Cost7049
\[\begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-8} \lor \neg \left(y \leq 4.4\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
Alternative 8
Accuracy54.6%
Cost6984
\[\begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{+199}:\\ \;\;\;\;y \cdot \left(-x \cdot t\right)\\ \mathbf{elif}\;t \leq -1 \cdot 10^{+50}:\\ \;\;\;\;x \cdot e^{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 9
Accuracy53.5%
Cost6920
\[\begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+233}:\\ \;\;\;\;y \cdot \left(-x \cdot t\right)\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{+38}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 10
Accuracy28.4%
Cost1484
\[\begin{array}{l} t_1 := a \cdot \left(x \cdot \left(-z\right)\right)\\ t_2 := x - y \cdot \left(x \cdot t\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-133}:\\ \;\;\;\;x \cdot \frac{1 - \left(y \cdot t\right) \cdot \left(y \cdot t\right)}{y \cdot t + 1}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+63} \lor \neg \left(x \leq 1.1 \cdot 10^{+165}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(a \cdot b\right)\\ \end{array} \]
Alternative 11
Accuracy27.7%
Cost1241
\[\begin{array}{l} t_1 := a \cdot \left(x \cdot \left(-z\right)\right)\\ t_2 := x - y \cdot \left(x \cdot t\right)\\ \mathbf{if}\;x \leq -6 \cdot 10^{-122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-148}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+63} \lor \neg \left(x \leq 3.05 \cdot 10^{+165}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(a \cdot b\right)\\ \end{array} \]
Alternative 12
Accuracy22.2%
Cost912
\[\begin{array}{l} t_1 := a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-122}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 10^{-147}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \]
Alternative 13
Accuracy22.4%
Cost912
\[\begin{array}{l} t_1 := a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-123}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-148}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-x \cdot t\right)\\ \end{array} \]
Alternative 14
Accuracy29.6%
Cost844
\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{+236}:\\ \;\;\;\;y \cdot \left(-x \cdot t\right)\\ \mathbf{elif}\;t \leq -1.96 \cdot 10^{+37}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-19}:\\ \;\;\;\;x - a \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
Alternative 15
Accuracy29.6%
Cost844
\[\begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{+234}:\\ \;\;\;\;y \cdot \left(-x \cdot t\right)\\ \mathbf{elif}\;t \leq -8.4 \cdot 10^{+38}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-16}:\\ \;\;\;\;x - a \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(y \cdot t\right)\\ \end{array} \]
Alternative 16
Accuracy30.1%
Cost712
\[\begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{+198}:\\ \;\;\;\;b \cdot \left(-x \cdot a\right)\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+95}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
Alternative 17
Accuracy23.0%
Cost516
\[\begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-178}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
Alternative 18
Accuracy21.0%
Cost452
\[\begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]
Alternative 19
Accuracy21.1%
Cost452
\[\begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \]
Alternative 20
Accuracy19.6%
Cost64
\[x \]

Reproduce?

herbie shell --seed 2023165 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))