Average Error: 0.1 → 0.1
Time: 5.9s
Precision: binary64
\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
↓
\[\left(z + \left(\left(x + y\right) - z \cdot \log t\right)\right) + \mathsf{fma}\left(a, b, b \cdot -0.5\right) \]
(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
 (+ (+ z (- (+ x y) (* z (log t)))) (fma a b (* b -0.5))))
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 (z + ((x + y) - (z * log(t)))) + fma(a, b, (b * -0.5));
}

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(z + Float64(Float64(x + y) - Float64(z * log(t)))) + fma(a, b, Float64(b * -0.5)))
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[(z + N[(N[(x + y), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * b + N[(b * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b
↓
\left(z + \left(\left(x + y\right) - z \cdot \log t\right)\right) + \mathsf{fma}\left(a, b, b \cdot -0.5\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. Applied egg-rr0.1

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

  3. Applied egg-rr0.1

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

  4. Applied egg-rr0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022153 
(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)))