?

Average Error: 2.98% → 0.36%
Time: 1.4min
Precision: binary64

?

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

Error?

Derivation?

  1. Initial program 2.98

    \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
  2. Simplified0.36

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

Reproduce?

herbie shell --seed 2023136 
(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))))))