?

Average Error: 3.16% → 0.48%
Time: 24.1s
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 3.16

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

  2. Simplified0.48

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

    [Start]3.16

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

    fma-def [=>]2.82

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

    sub-neg [=>]2.82

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

    log1p-def [=>]0.48

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

  3. Final simplification0.48

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

Alternatives

Alternative 1
Error3.11%
Cost13892
\[\begin{array}{l} \mathbf{if}\;a \leq 1.8 \cdot 10^{+73}:\\ \;\;\;\;x \cdot e^{\left(1 - a \cdot b\right) + \left(-1 + y \cdot \left(\log z - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-\left(z + b\right)\right)}\\ \end{array} \]
Alternative 2
Error11.64%
Cost13772
\[\begin{array}{l} t_1 := x \cdot e^{y \cdot \log z - a \cdot b}\\ \mathbf{if}\;a \leq -240000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.56 \cdot 10^{-131}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-\left(z + b\right)\right)}\\ \end{array} \]
Alternative 3
Error11.68%
Cost13772
\[\begin{array}{l} t_1 := x \cdot e^{y \cdot \log z - a \cdot b}\\ \mathbf{if}\;a \leq -2800000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.56 \cdot 10^{-131}:\\ \;\;\;\;x \cdot e^{1 + \left(-1 + y \cdot \left(\log z - t\right)\right)}\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-\left(z + b\right)\right)}\\ \end{array} \]
Alternative 4
Error3.11%
Cost13636
\[\begin{array}{l} \mathbf{if}\;a \leq 1.8 \cdot 10^{+73}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-\left(z + b\right)\right)}\\ \end{array} \]
Alternative 5
Error10.14%
Cost13512
\[\begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-15}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-12}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-\left(z + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
Alternative 6
Error9.49%
Cost7176
\[\begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-15}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-8}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-\left(z + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 7
Error12.1%
Cost7048
\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-15}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 8
Error16.33%
Cost6916
\[\begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 9
Error16.33%
Cost6852
\[\begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 10
Error39.85%
Cost6788
\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-13}:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 11
Error61.93%
Cost712
\[\begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-13}:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;y \leq 6600000:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \end{array} \]
Alternative 12
Error61.93%
Cost712
\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-13}:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;y \leq 6600000:\\ \;\;\;\;x - x \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \end{array} \]
Alternative 13
Error62.05%
Cost585
\[\begin{array}{l} \mathbf{if}\;y \leq -30000000000000 \lor \neg \left(y \leq 6600000\right):\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 14
Error61.96%
Cost584
\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-13}:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;y \leq 6600000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \end{array} \]
Alternative 15
Error69.52%
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023088 
(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))))))