Average Error: 1.7 → 0.2
Time: 17.3s
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 1.7

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

  2. Simplified0.2

    \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
    Proof
    (*.f64 x (exp.f64 (fma.f64 y (-.f64 (log.f64 z) t) (*.f64 a (-.f64 (log1p.f64 (neg.f64 z)) b))))): 0 points increase in error, 0 points decrease in error
    (*.f64 x (exp.f64 (fma.f64 y (-.f64 (log.f64 z) t) (*.f64 a (-.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (neg.f64 z)))) b))))): 0 points increase in error, 0 points decrease in error
    (*.f64 x (exp.f64 (fma.f64 y (-.f64 (log.f64 z) t) (*.f64 a (-.f64 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 z))) b))))): 0 points increase in error, 0 points decrease in error
    (*.f64 x (exp.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 1 z)) b)))))): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.2

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

Alternatives

Alternative 1
Error0.6
Cost33860
\[\begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-16}:\\ \;\;\;\;x \cdot e^{t_1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \]
Alternative 2
Error25.9
Cost7184
\[\begin{array}{l} t_1 := x \cdot {z}^{y}\\ t_2 := x + a \cdot \left(x \cdot b\right)\\ t_3 := \frac{x \cdot x}{t_2}\\ \mathbf{if}\;b \leq -3 \cdot 10^{+185}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -1.25 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-167}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot \left(-x \cdot x\right)\right)}{t_2}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+146}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 3
Error6.4
Cost7176
\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-56}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 4
Error24.2
Cost7052
\[\begin{array}{l} t_1 := \frac{x \cdot x}{x + a \cdot \left(x \cdot b\right)}\\ \mathbf{if}\;y \leq 9.4 \cdot 10^{-282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-103}:\\ \;\;\;\;x \cdot e^{a \cdot b}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 5
Error7.8
Cost7048
\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-58}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 6
Error10.0
Cost6916
\[\begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 7
Error33.5
Cost1676
\[\begin{array}{l} t_1 := x + a \cdot \left(x \cdot b\right)\\ t_2 := \frac{x \cdot x}{t_1}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;x + x \cdot \left(\left(a \cdot b\right) \cdot \left(-1 + \left(a \cdot b\right) \cdot 0.5\right)\right)\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{+18}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-43}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot \left(-x \cdot x\right)\right)}{t_1}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 8
Error32.6
Cost1092
\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;x + x \cdot \left(\left(a \cdot b\right) \cdot \left(-1 + \left(a \cdot b\right) \cdot 0.5\right)\right)\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot x}{x + a \cdot \left(x \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 9
Error32.6
Cost968
\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot x}{x + a \cdot \left(x \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 10
Error39.4
Cost648
\[\begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+19}:\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \]
Alternative 11
Error39.4
Cost648
\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+18}:\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \end{array} \]
Alternative 12
Error39.5
Cost585
\[\begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{+21} \lor \neg \left(y \leq 6.8 \cdot 10^{-18}\right):\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 13
Error44.2
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022343 
(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))))))