Example from Robby

Average Accuracy: 99.8% → 99.8%

Time: 23.4s

Precision: binary64

Cost: 58880

Math

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

↓

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

FPCore

(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
 (fabs
  (fma
   (* ew (sin t))
   (/ 1.0 (hypot 1.0 (/ (/ eh ew) (tan t))))
   (* eh (* (cos t) (sin (atan (/ eh (* ew (tan t))))))))))

C

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) {
	return fabs(fma((ew * sin(t)), (1.0 / hypot(1.0, ((eh / ew) / tan(t)))), (eh * (cos(t) * sin(atan((eh / (ew * tan(t)))))))));
}

Julia

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)
	return abs(fma(Float64(ew * sin(t)), Float64(1.0 / hypot(1.0, Float64(Float64(eh / ew) / tan(t)))), Float64(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * tan(t)))))))))
end

Wolfram

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_] := N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Simplified 99.8%

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

   [Start] 99.8	\[ \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   fma-def [=>] 99.8	\[ \left|\color{blue}{\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)}\right| \]
   associate-/l/ [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \color{blue}{\left(\frac{eh}{\tan t \cdot ew}\right)}, \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right| \]
   associate-*l* [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)}\right)\right| \]
   associate-/l/ [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{\tan t \cdot ew}\right)}\right)\right)\right| \]

3. Applied egg-rr 99.8%

   \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{{\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{-0.5}}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   Step-by-step derivation

   [Start] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   add-sqr-sqrt [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{\sqrt{\cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)} \cdot \sqrt{\cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   pow1/2 [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{{\cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}^{0.5}} \cdot \sqrt{\cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   cos-atan [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, {\color{blue}{\left(\frac{1}{\sqrt{1 + \frac{eh}{\tan t \cdot ew} \cdot \frac{eh}{\tan t \cdot ew}}}\right)}}^{0.5} \cdot \sqrt{\cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   inv-pow [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, {\color{blue}{\left({\left(\sqrt{1 + \frac{eh}{\tan t \cdot ew} \cdot \frac{eh}{\tan t \cdot ew}}\right)}^{-1}\right)}}^{0.5} \cdot \sqrt{\cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   pow-pow [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{{\left(\sqrt{1 + \frac{eh}{\tan t \cdot ew} \cdot \frac{eh}{\tan t \cdot ew}}\right)}^{\left(-1 \cdot 0.5\right)}} \cdot \sqrt{\cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   hypot-1-def [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, {\color{blue}{\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}}^{\left(-1 \cdot 0.5\right)} \cdot \sqrt{\cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   metadata-eval [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, {\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{\color{blue}{-0.5}} \cdot \sqrt{\cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   pow1/2 [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, {\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{-0.5} \cdot \color{blue}{{\cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}^{0.5}}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   cos-atan [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, {\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{-0.5} \cdot {\color{blue}{\left(\frac{1}{\sqrt{1 + \frac{eh}{\tan t \cdot ew} \cdot \frac{eh}{\tan t \cdot ew}}}\right)}}^{0.5}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   inv-pow [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, {\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{-0.5} \cdot {\color{blue}{\left({\left(\sqrt{1 + \frac{eh}{\tan t \cdot ew} \cdot \frac{eh}{\tan t \cdot ew}}\right)}^{-1}\right)}}^{0.5}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   pow-pow [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, {\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{-0.5} \cdot \color{blue}{{\left(\sqrt{1 + \frac{eh}{\tan t \cdot ew} \cdot \frac{eh}{\tan t \cdot ew}}\right)}^{\left(-1 \cdot 0.5\right)}}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   hypot-1-def [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, {\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{-0.5} \cdot {\color{blue}{\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}}^{\left(-1 \cdot 0.5\right)}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   metadata-eval [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, {\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{\color{blue}{-0.5}}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]

4. Simplified 99.8%

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

   [Start] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, {\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{-0.5}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   pow-sqr [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{{\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{\left(2 \cdot -0.5\right)}}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   metadata-eval [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, {\left(\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)\right)}^{\color{blue}{-1}}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   unpow-1 [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{\frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)}}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   *-commutative [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\color{blue}{ew \cdot \tan t}}\right)}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]
   associate-/r* [=>] 99.8	\[ \left|\mathsf{fma}\left(ew \cdot \sin t, \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{\frac{eh}{ew}}{\tan t}}\right)}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right| \]

5. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	52608

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

Alternative 2
Accuracy	99.0%
Cost	52416

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

Alternative 3
Accuracy	98.4%
Cost	39232

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

Alternative 4
Accuracy	97.9%
Cost	19648

\[\left|ew \cdot \sin t + eh \cdot \cos t\right| \]

Alternative 5
Accuracy	97.9%
Cost	19648

\[\left|ew \cdot \sin t - eh \cdot \cos t\right| \]

Alternative 6
Accuracy	78.5%
Cost	13120

\[\left|eh + ew \cdot \sin t\right| \]

Alternative 7
Accuracy	42.3%
Cost	12992

\[\left|ew \cdot \sin t\right| \]

Reproduce

herbie shell --seed 2023157 
(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))))))))