Average Error: 2.0 → 2.0
Time: 29.5s
Precision: binary64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
↓
\[\frac{x \cdot e^{\mathsf{fma}\left(\log a, t + -1, \log z \cdot y\right) - b}}{y} \]
(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))
↓
(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (fma (log a) (+ t -1.0) (* (log z) y)) b))) y))
double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((fma(log(a), (t + -1.0), (log(z) * y)) - b))) / y;
}

function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
end
↓
function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(fma(log(a), Float64(t + -1.0), Float64(log(z) * y)) - b))) / y)
end

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original2.0
Target11.1
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array} \]

Derivation

  1. Initial program 2.0

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

  2. Applied add-exp-log_binary642.0

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

  3. Simplified2.0

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

  4. Final simplification2.0

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

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))