Average Error: 0.1 → 0.1
Time: 9.4s
Precision: binary64
\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
↓
\[\mathsf{fma}\left(\log \left(\sqrt{t}\right), -2, 1\right) \cdot z + \mathsf{fma}\left(a - 0.5, b, x + y\right) \]
(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
 (+ (* (fma (log (sqrt t)) -2.0 1.0) z) (fma (- a 0.5) b (+ x y))))
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 (fma(log(sqrt(t)), -2.0, 1.0) * z) + fma((a - 0.5), b, (x + y));
}

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(fma(log(sqrt(t)), -2.0, 1.0) * z) + fma(Float64(a - 0.5), b, Float64(x + y)))
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[(N[Log[N[Sqrt[t], $MachinePrecision]], $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] * z), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b
↓
\mathsf{fma}\left(\log \left(\sqrt{t}\right), -2, 1\right) \cdot z + \mathsf{fma}\left(a - 0.5, b, x + y\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
Target0.3
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

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

  3. Applied fma-udef_binary640.1

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

  4. Simplified0.1

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

  5. Applied add-sqr-sqrt_binary640.1

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

  6. Applied log-prod_binary640.1

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

  7. Applied distribute-rgt-in_binary640.1

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

  8. Applied associate--r+_binary640.1

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

  9. Applied add-cube-cbrt_binary640.2

    \[\leadsto \left(\color{blue}{\left(\sqrt[3]{z - \log \left(\sqrt{t}\right) \cdot z} \cdot \sqrt[3]{z - \log \left(\sqrt{t}\right) \cdot z}\right) \cdot \sqrt[3]{z - \log \left(\sqrt{t}\right) \cdot z}} - \log \left(\sqrt{t}\right) \cdot z\right) + \mathsf{fma}\left(a - 0.5, b, x + y\right) \]

  10. Applied prod-diff_binary640.2

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{z - \log \left(\sqrt{t}\right) \cdot z} \cdot \sqrt[3]{z - \log \left(\sqrt{t}\right) \cdot z}, \sqrt[3]{z - \log \left(\sqrt{t}\right) \cdot z}, -z \cdot \log \left(\sqrt{t}\right)\right) + \mathsf{fma}\left(-z, \log \left(\sqrt{t}\right), z \cdot \log \left(\sqrt{t}\right)\right)\right)} + \mathsf{fma}\left(a - 0.5, b, x + y\right) \]

  11. Simplified0.1

    \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{t}\right), -2, 1\right) \cdot z} + \mathsf{fma}\left(-z, \log \left(\sqrt{t}\right), z \cdot \log \left(\sqrt{t}\right)\right)\right) + \mathsf{fma}\left(a - 0.5, b, x + y\right) \]

  12. Simplified0.1

    \[\leadsto \left(\mathsf{fma}\left(\log \left(\sqrt{t}\right), -2, 1\right) \cdot z + \color{blue}{0}\right) + \mathsf{fma}\left(a - 0.5, b, x + y\right) \]

  13. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\log \left(\sqrt{t}\right), -2, 1\right) \cdot z + \mathsf{fma}\left(a - 0.5, b, x + y\right) \]

Reproduce

herbie shell --seed 2022131 
(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)))