Example from Robby

Average Accuracy: 99.8% → 99.8%

Time: 21.3s

Precision: binary64

Cost: 58880

Math

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

↓

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

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
 (let* ((t_1 (/ eh (* ew (tan t)))))
   (fabs
    (fma
     (* ew (sin t))
     (/ 1.0 (hypot 1.0 t_1))
     (* eh (* (cos t) (sin (atan t_1))))))))

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) {
	double t_1 = eh / (ew * tan(t));
	return fabs(fma((ew * sin(t)), (1.0 / hypot(1.0, t_1)), (eh * (cos(t) * sin(atan(t_1))))));
}

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)
	t_1 = Float64(eh / Float64(ew * tan(t)))
	return abs(fma(Float64(ew * sin(t)), Float64(1.0 / hypot(1.0, t_1)), Float64(eh * Float64(cos(t) * sin(atan(t_1))))))
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_] := Block[{t$95$1 = N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] + N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $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%

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

   [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%

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

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	58816

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

Alternative 2
Accuracy	99.7%
Cost	52480

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

Alternative 3
Accuracy	99.7%
Cost	52480

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

Alternative 4
Accuracy	99.1%
Cost	52416

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

Alternative 5
Accuracy	99.0%
Cost	46080

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

Alternative 6
Accuracy	99.0%
Cost	46080

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

Alternative 7
Accuracy	98.6%
Cost	39232

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

Alternative 8
Accuracy	96.7%
Cost	33737

\[\begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;ew \leq -1.36 \cdot 10^{-159} \lor \neg \left(ew \leq 4.8 \cdot 10^{-159}\right):\\ \;\;\;\;\left|ew \cdot \sin t + t_1 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t} + -0.3333333333333333 \cdot \frac{t \cdot eh}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \frac{1}{0.16666666666666666 \cdot \frac{t}{ew} + \frac{1}{ew \cdot t}}\right|\\ \end{array} \]

Alternative 9
Accuracy	96.5%
Cost	33609

\[\begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;ew \leq -3.6 \cdot 10^{-160} \lor \neg \left(ew \leq 1.56 \cdot 10^{-161}\right):\\ \;\;\;\;\left|ew \cdot \sin t + t_1 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t} + -0.3333333333333333 \cdot \frac{t \cdot eh}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + t \cdot \frac{t \cdot \left(ew \cdot ew\right)}{eh}\right|\\ \end{array} \]

Alternative 10
Accuracy	90.4%
Cost	33481

\[\begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;ew \leq -3.6 \cdot 10^{-160} \lor \neg \left(ew \leq 10^{-250}\right):\\ \;\;\;\;\left|ew \cdot \sin t + t_1 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + t \cdot \frac{t \cdot \left(ew \cdot ew\right)}{eh}\right|\\ \end{array} \]

Alternative 11
Accuracy	83.5%
Cost	33097

\[\begin{array}{l} \mathbf{if}\;ew \leq -2.05 \cdot 10^{-91} \lor \neg \left(ew \leq 4.9 \cdot 10^{-118}\right):\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + ew \cdot t\right|\\ \end{array} \]

Alternative 12
Accuracy	83.0%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;ew \leq -3.4 \cdot 10^{-92} \lor \neg \left(ew \leq 3.75 \cdot 10^{-118}\right):\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot \frac{t \cdot \left(ew \cdot ew\right)}{eh} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \]

Alternative 13
Accuracy	89.3%
Cost	32832

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

Alternative 14
Accuracy	50.6%
Cost	27460

\[\begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ t_2 := t \cdot \frac{t \cdot \left(ew \cdot ew\right)}{eh}\\ t_3 := eh \cdot \cos t\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;\left|t_3 \cdot \sin \tan^{-1} \left(t_1 + -0.3333333333333333 \cdot \frac{t \cdot eh}{ew}\right) + t_2\right|\\ \mathbf{elif}\;t \leq 0.0116:\\ \;\;\;\;\left|\left(ew \cdot t\right) \cdot \frac{ew}{\frac{eh}{t}} + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_2 + t_3 \cdot \sin \tan^{-1} t_1\right|\\ \end{array} \]

Alternative 15
Accuracy	50.2%
Cost	27081

\[\begin{array}{l} \mathbf{if}\;t \leq -0.0005 \lor \neg \left(t \leq 0.0116\right):\\ \;\;\;\;\left|t \cdot \frac{t \cdot \left(ew \cdot ew\right)}{eh} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot t\right) \cdot \frac{ew}{\frac{eh}{t}} + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \end{array} \]

Alternative 16
Accuracy	38.6%
Cost	26688

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

Alternative 17
Accuracy	39.2%
Cost	26688

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

Alternative 18
Accuracy	33.8%
Cost	20288

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

Reproduce

herbie shell --seed 2023129 
(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))))))))