?

Average Error: 2.2 → 0.3
Time: 22.2s
Precision: binary64
Cost: 26368

?

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

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(y, Float64(log(z) - t), Float64(a * Float64(log1p(Float64(-z)) - b)))))
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[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[(N[Log[1 + (-z)], $MachinePrecision] - b), $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(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}

Error?

Derivation?

  1. Initial program 2.2

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

  2. Simplified0.3

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

    [Start]2.2

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

    fma-def [=>]2.0

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

    sub-neg [=>]2.0

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

    log1p-def [=>]0.3

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

  3. Final simplification0.3

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

Alternatives

Alternative 1
Error10.0
Cost13708
\[\begin{array}{l} t_1 := e^{a \cdot b}\\ \mathbf{if}\;a \leq -2.9 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-162}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \end{array} \]
Alternative 2
Error8.5
Cost13708
\[\begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{-53}:\\ \;\;\;\;x \cdot e^{y \cdot \log z - a \cdot b}\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-162}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+111}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{e^{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \end{array} \]
Alternative 3
Error1.6
Cost13636
\[\begin{array}{l} \mathbf{if}\;a \leq 1.12 \cdot 10^{+122}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \end{array} \]
Alternative 4
Error7.0
Cost13576
\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-81}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;y \leq 0.68:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 5
Error8.6
Cost13380
\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-81}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 6
Error22.2
Cost7316
\[\begin{array}{l} t_1 := x - t \cdot \left(x \cdot y\right)\\ t_2 := \left(x + 1\right) + -1\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{-206}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-223}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-40}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 7
Error22.2
Cost7316
\[\begin{array}{l} t_1 := \left(x + 1\right) + -1\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-255}:\\ \;\;\;\;x \cdot e^{a \cdot b}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{-89}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 8
Error8.4
Cost6984
\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-81}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq 0.0105:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 9
Error10.9
Cost6852
\[\begin{array}{l} \mathbf{if}\;y \leq 3.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 10
Error31.6
Cost1753
\[\begin{array}{l} t_1 := x - t \cdot \left(x \cdot y\right)\\ t_2 := \left(x + 1\right) + -1\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{-206}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-220}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-40} \lor \neg \left(y \leq 3.4 \cdot 10^{-13}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(a \cdot b\right) \cdot \left(-1 + \left(a \cdot b\right) \cdot 0.5\right)\right)\\ \end{array} \]
Alternative 11
Error31.5
Cost1753
\[\begin{array}{l} t_1 := x - t \cdot \left(x \cdot y\right)\\ t_2 := \left(x + 1\right) + -1\\ \mathbf{if}\;y \leq -1.95 \cdot 10^{-206}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-40} \lor \neg \left(y \leq 7 \cdot 10^{-13}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(\left(a \cdot b\right) \cdot \left(-1 + \left(a \cdot b\right) \cdot 0.5\right)\right)\\ \end{array} \]
Alternative 12
Error31.4
Cost978
\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-201} \lor \neg \left(y \leq 7.2 \cdot 10^{-255} \lor \neg \left(y \leq 1.75 \cdot 10^{-221}\right) \land y \leq 6.6 \cdot 10^{-88}\right):\\ \;\;\;\;\left(x + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
Alternative 13
Error31.4
Cost978
\[\begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-206} \lor \neg \left(y \leq 1.65 \cdot 10^{-259}\right) \land \left(y \leq 2.85 \cdot 10^{-221} \lor \neg \left(y \leq 2.5 \cdot 10^{-87}\right)\right):\\ \;\;\;\;\left(x + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \end{array} \]
Alternative 14
Error31.4
Cost850
\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-198} \lor \neg \left(y \leq 2.8 \cdot 10^{-257}\right) \land \left(y \leq 1.5 \cdot 10^{-222} \lor \neg \left(y \leq 7.5 \cdot 10^{-93}\right)\right):\\ \;\;\;\;\left(x + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 15
Error44.4
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023060 
(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))))))