Example from Robby

Average Error: 0.1 → 0.1

Time: 18.7s

Precision: binary64

Cost: 58816

Math

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

↓

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

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 (+ (* (cos t_1) (* ew (sin t))) (* (* 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(((cos(t_1) * (ew * sin(t))) + ((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(((cos(t_1) * (ew * sin(t))) + ((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(((Math.cos(t_1) * (ew * Math.sin(t))) + ((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(((math.cos(t_1) * (ew * math.sin(t))) + ((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(cos(t_1) * Float64(ew * sin(t))) + 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(((cos(t_1) * (ew * sin(t))) + ((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[Cos[t$95$1], $MachinePrecision] * N[(ew * N[Sin[t], $MachinePrecision]), $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

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

Alternatives

Alternative 1
Error	0.2
Cost	52480

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

Alternative 2
Error	0.7
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 3
Error	1.0
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 4
Error	2.2
Cost	33609

\[\begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;ew \leq -5.1 \cdot 10^{-155} \lor \neg \left(ew \leq 1.32 \cdot 10^{-222}\right):\\ \;\;\;\;\left|ew \cdot \sin t + t_1 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t} + -0.3333333333333333 \cdot \frac{t \cdot eh}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \frac{ew}{t \cdot 0.16666666666666666 + \frac{1}{t}}\right|\\ \end{array} \]

Alternative 5
Error	6.9
Cost	33348

\[\begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq 650000000000:\\ \;\;\;\;\left|ew \cdot \sin t + t_1 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \frac{ew}{t \cdot 0.16666666666666666 + \frac{1}{t}}\right|\\ \end{array} \]

Alternative 6
Error	6.8
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 7
Error	8.9
Cost	27145

\[\begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ \mathbf{if}\;t \leq -220000000 \lor \neg \left(t \leq 5.4 \cdot 10^{+20}\right):\\ \;\;\;\;\left|ew \cdot \sin t + \left(\cos t \cdot t_1\right) \cdot \frac{eh}{\mathsf{hypot}\left(1, t_1\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} t_1\right|\\ \end{array} \]

Alternative 8
Error	8.9
Cost	27144

\[\begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := eh \cdot \cos t\\ t_3 := \frac{eh}{ew \cdot t}\\ t_4 := \mathsf{hypot}\left(1, t_3\right)\\ \mathbf{if}\;t \leq -220000000:\\ \;\;\;\;\left|t_1 + \left(\cos t \cdot t_3\right) \cdot \frac{eh}{t_4}\right|\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+20}:\\ \;\;\;\;\left|ew \cdot t + t_2 \cdot \sin \tan^{-1} t_3\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 + \frac{t_2}{t_4 \cdot \left(ew \cdot \frac{t}{eh}\right)}\right|\\ \end{array} \]

Alternative 9
Error	11.0
Cost	26697

\[\begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\\ \mathbf{if}\;eh \leq -1.3 \cdot 10^{+111} \lor \neg \left(eh \leq 2.8 \cdot 10^{+52}\right):\\ \;\;\;\;\left|ew \cdot t + \left(eh \cdot \cos t\right) \cdot t_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot t_1\right|\\ \end{array} \]

Alternative 10
Error	13.8
Cost	26304

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

Alternative 11
Error	13.8
Cost	26304

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

Alternative 12
Error	42.7
Cost	20288

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

Reproduce

herbie shell --seed 2023066 
(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))))))))