Average Error: 0.1 → 0.2
Time: 13.1s
Precision: binary64
\[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
↓
\[\begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{\tan t}\\ \left|\mathsf{fma}\left(ew, \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin t}{\mathsf{hypot}\left(1, t_1\right)}\right)\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} t_1\right)\right)\right| \end{array} \]
(FPCore (eh ew t)
 :precision binary64
 (fabs
  (+
   (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t)))))
   (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))
↓
(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) (tan t))))
   (fabs
    (fma
     ew
     (expm1 (log1p (/ (sin t) (hypot 1.0 t_1))))
     (* eh (* (cos t) (sin (atan t_1))))))))
double code(double eh, double ew, double t) {
	return fabs((((ew * sin(t)) * cos(atan(((eh / ew) / tan(t))))) + ((eh * cos(t)) * sin(atan(((eh / ew) / tan(t)))))));
}
↓
double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / tan(t);
	return fabs(fma(ew, expm1(log1p((sin(t) / hypot(1.0, t_1)))), (eh * (cos(t) * sin(atan(t_1))))));
}

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(ew * sin(t)) * cos(atan(Float64(Float64(eh / ew) / tan(t))))) + Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / ew) / tan(t)))))))
end
↓
function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / tan(t))
	return abs(fma(ew, expm1(log1p(Float64(sin(t) / hypot(1.0, t_1)))), Float64(eh * Float64(cos(t) * sin(atan(t_1))))))
end

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(ew * N[(Exp[N[Log[1 + N[(N[Sin[t], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] + N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|
↓
\begin{array}{l}
t_1 := \frac{\frac{eh}{ew}}{\tan t}\\
\left|\mathsf{fma}\left(ew, \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin t}{\mathsf{hypot}\left(1, t_1\right)}\right)\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} t_1\right)\right)\right|
\end{array}

Error

Bits error versus eh

Bits error versus ew

Bits error versus t

Derivation

  1. Initial program 0.1

    \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right|} \]

  3. Applied egg-rr0.6

    \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{{\left(\sqrt[3]{\frac{\sin t}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}}\right)}^{3}}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right| \]

  4. Applied egg-rr0.2

    \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin t}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}\right)\right)}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right| \]

  5. Final simplification0.2

    \[\leadsto \left|\mathsf{fma}\left(ew, \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin t}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}\right)\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right| \]

Reproduce

herbie shell --seed 2022153 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))