Average Error: 2.2 → 0.3
Time: 26.1s
Precision: binary64
Cost: 26368
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
↓
\[x \cdot e^{\mathsf{fma}\left(a, \mathsf{log1p}\left(-z\right) - b, y \cdot \left(\log z - t\right)\right)} \]
(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)}

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

    \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(a, \mathsf{log1p}\left(-z\right) - b, y \cdot \left(\log z - t\right)\right)}} \]
    Proof
    (*.f64 x (exp.f64 (fma.f64 a (-.f64 (log1p.f64 (neg.f64 z)) b) (*.f64 y (-.f64 (log.f64 z) t))))): 0 points increase in error, 0 points decrease in error
    (*.f64 x (exp.f64 (fma.f64 a (-.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (neg.f64 z)))) b) (*.f64 y (-.f64 (log.f64 z) t))))): 5 points increase in error, 0 points decrease in error
    (*.f64 x (exp.f64 (fma.f64 a (-.f64 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 z))) b) (*.f64 y (-.f64 (log.f64 z) t))))): 0 points increase in error, 0 points decrease in error
    (*.f64 x (exp.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (-.f64 (log.f64 (-.f64 1 z)) b)) (*.f64 y (-.f64 (log.f64 z) t)))))): 3 points increase in error, 0 points decrease in error
    (*.f64 x (exp.f64 (Rewrite<= +-commutative_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 simplification0.3

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

Alternatives

Alternative 1
Error6.4
Cost13640
\[\begin{array}{l} t_1 := x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+169}:\\ \;\;\;\;x \cdot e^{y \cdot \log z - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error6.2
Cost13576
\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+57}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq 0.00049:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 3
Error2.4
Cost13504
\[x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot b} \]
Alternative 4
Error8.1
Cost13448
\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-31}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq 0.00049:\\ \;\;\;\;x \cdot {e}^{\left(a \cdot \left(-b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 5
Error8.1
Cost7048
\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-31}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq 0.00049:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 6
Error15.5
Cost6916
\[\begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 7
Error23.6
Cost6788
\[\begin{array}{l} \mathbf{if}\;y \leq -0.6:\\ \;\;\;\;\left(1 + b \cdot \left(x \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
Alternative 8
Error32.5
Cost840
\[\begin{array}{l} t_1 := \left(1 + b \cdot \left(x \cdot a\right)\right) + -1\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 390:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 9
Error39.9
Cost648
\[\begin{array}{l} t_1 := a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 390:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error39.6
Cost648
\[\begin{array}{l} t_1 := b \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{if}\;y \leq -4400000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 390:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 11
Error44.5
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022308 
(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))))))