Example from Robby

Average Error: 0.1 → 0.1

Time: 18.2s

Precision: binary64

Cost: 58816

Math

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

↓

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

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 (atan (/ (/ eh ew) (tan t)))))
   (fabs (+ (* (* ew (sin t)) (cos t_1)) (* (* eh (cos t)) (sin 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 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((((ew * sin(t)) * cos(atan(((eh / ew) / tan(t))))) + ((eh * cos(t)) * sin(atan(((eh / ew) / tan(t)))))))
end function

↓

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((eh / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((ew * Math.sin(t)) * Math.cos(Math.atan(((eh / ew) / Math.tan(t))))) + ((eh * Math.cos(t)) * Math.sin(Math.atan(((eh / ew) / Math.tan(t)))))));
}

↓

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh / ew) / Math.tan(t)));
	return Math.abs((((ew * Math.sin(t)) * Math.cos(t_1)) + ((eh * Math.cos(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	return math.fabs((((ew * math.sin(t)) * math.cos(math.atan(((eh / ew) / math.tan(t))))) + ((eh * math.cos(t)) * math.sin(math.atan(((eh / ew) / math.tan(t)))))))

↓

def code(eh, ew, t):
	t_1 = math.atan(((eh / ew) / math.tan(t)))
	return math.fabs((((ew * math.sin(t)) * math.cos(t_1)) + ((eh * math.cos(t)) * math.sin(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 = atan(Float64(Float64(eh / ew) / tan(t)))
	return abs(Float64(Float64(Float64(ew * sin(t)) * cos(t_1)) + Float64(Float64(eh * cos(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((ew * sin(t)) * cos(atan(((eh / ew) / tan(t))))) + ((eh * cos(t)) * sin(atan(((eh / ew) / tan(t)))))));
end

↓

function tmp = code(eh, ew, t)
	t_1 = atan(((eh / ew) / tan(t)));
	tmp = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(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[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

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. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.2
Cost	52480

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

Alternative 2
Error	0.1
Cost	52480

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

Alternative 3
Error	0.6
Cost	52416

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

Alternative 4
Error	0.9
Cost	39232

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

Alternative 5
Error	16.0
Cost	33496

\[\begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{\tan t}\\ t_2 := ew \cdot \sin t\\ t_3 := \left|t_2 + eh \cdot \sin \tan^{-1} t_1\right|\\ t_4 := \left|eh + \frac{t_2}{\mathsf{hypot}\left(1, t_1\right)}\right|\\ t_5 := \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\\ t_6 := \left|t_5 + \left(ew \cdot ew\right) \cdot \frac{t \cdot t}{eh}\right|\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{+178}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{+160}:\\ \;\;\;\;\left|\left(ew \cdot ew\right) \cdot \left(\left(1 + \frac{t}{\frac{eh}{t}}\right) + -1\right) + t_5\right|\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{+132}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{+29}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+46}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+132}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Error	10.8
Cost	33480

\[\begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ t_2 := \left|\left(eh \cdot \cos t\right) \cdot t_1 + ew \cdot \left(\left(ew \cdot t\right) \cdot \frac{t}{eh}\right)\right|\\ \mathbf{if}\;eh \leq -9.5 \cdot 10^{+144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;eh \leq 7.5 \cdot 10^{+86}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot t_1\right|\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	8.6
Cost	33480

\[\begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ t_2 := \left|ew \cdot \sin t + eh \cdot t_1\right|\\ \mathbf{if}\;ew \leq -2 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;ew \leq 1.3 \cdot 10^{-130}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot t_1 + \frac{t}{\frac{\frac{eh}{ew \cdot ew}}{t}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	16.1
Cost	27608

\[\begin{array}{l} t_1 := \frac{ew \cdot \sin t}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}\\ t_2 := \left|eh + t_1\right|\\ t_3 := \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\\ t_4 := \left|t_3 + \left(ew \cdot ew\right) \cdot \frac{t \cdot t}{eh}\right|\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{+178}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{+160}:\\ \;\;\;\;\left|\left(ew \cdot ew\right) \cdot \left(\left(1 + \frac{t}{\frac{eh}{t}}\right) + -1\right) + t_3\right|\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{+132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.82 \cdot 10^{+28}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+47}:\\ \;\;\;\;\left|t_1 - eh\right|\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+133}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	15.8
Cost	27344

\[\begin{array}{l} t_1 := \left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot ew\right) \cdot \frac{t \cdot t}{eh}\right|\\ t_2 := \frac{ew \cdot \sin t}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}\\ t_3 := \left|eh + t_2\right|\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{+132}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+48}:\\ \;\;\;\;\left|t_2 - eh\right|\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+132}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Error	13.1
Cost	26368

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

Alternative 11
Error	13.1
Cost	26368

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

Alternative 12
Error	42.9
Cost	20288

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

Reproduce

herbie shell --seed 2022330 
(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))))))))