Average Error: 0.1 → 0.1
Time: 8.6s
Precision: binary64
\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]
\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
↓
\[\left(\left(y + z \cdot \left(1 - \log t\right)\right) + \mathsf{fma}\left(-\log t, z, z \cdot \log t\right)\right) + \mathsf{fma}\left(a + -0.5, b, x\right) \]
(FPCore (x y z t a b)
 :precision binary64
 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))
↓
(FPCore (x y z t a b)
 :precision binary64
 (+
  (+ (+ y (* z (- 1.0 (log t)))) (fma (- (log t)) z (* z (log t))))
  (fma (+ a -0.5) b x)))
double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return ((y + (z * (1.0 - log(t)))) + fma(-log(t), z, (z * log(t)))) + fma((a + -0.5), b, x);
}

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b))
end
↓
function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(y + Float64(z * Float64(1.0 - log(t)))) + fma(Float64(-log(t)), z, Float64(z * log(t)))) + fma(Float64(a + -0.5), b, x))
end

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(y + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-N[Log[t], $MachinePrecision]) * z + N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * b + x), $MachinePrecision]), $MachinePrecision]

\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b
↓
\left(\left(y + z \cdot \left(1 - \log t\right)\right) + \mathsf{fma}\left(-\log t, z, z \cdot \log t\right)\right) + \mathsf{fma}\left(a + -0.5, b, x\right)

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

Original0.1
Target0.4
Herbie0.1
\[\left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b \]

Derivation

  1. Initial program 0.1

    \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(z, 1 - \log t, y\right) + \mathsf{fma}\left(a + -0.5, b, x\right)} \]

  3. Taylor expanded in z around 0 0.1

    \[\leadsto \color{blue}{\left(\left(y + z\right) - z \cdot \log t\right)} + \mathsf{fma}\left(a + -0.5, b, x\right) \]

  4. Applied egg-rr0.1

    \[\leadsto \color{blue}{\left(\left(y + z \cdot \left(1 - \log t\right)\right) + \mathsf{fma}\left(-\log t, z, z \cdot \log t\right)\right)} + \mathsf{fma}\left(a + -0.5, b, x\right) \]

  5. Final simplification0.1

    \[\leadsto \left(\left(y + z \cdot \left(1 - \log t\right)\right) + \mathsf{fma}\left(-\log t, z, z \cdot \log t\right)\right) + \mathsf{fma}\left(a + -0.5, b, x\right) \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t)))) (* (- a 0.5) b))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))