Example from Robby

Average Error: 0.1 → 0.1

Time: 18.5s

Precision: binary64

Cost: 58816

Math

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

↓

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

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

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))) + (sin(t_1) * (eh * cos(t)))))
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))) + (Math.sin(t_1) * (eh * Math.cos(t)))));
}

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))) + (math.sin(t_1) * (eh * math.cos(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)
	t_1 = atan(Float64(Float64(eh / ew) / tan(t)))
	return abs(Float64(Float64(cos(t_1) * Float64(ew * sin(t))) + Float64(sin(t_1) * Float64(eh * cos(t)))))
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))) + (sin(t_1) * (eh * cos(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_] := 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[Sin[t$95$1], $MachinePrecision] * N[(eh * N[Cos[t], $MachinePrecision]), $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

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

Alternatives

Alternative 1
Error	0.2
Cost	52480

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

Alternative 2
Error	0.5
Cost	52416

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

Alternative 3
Error	0.6
Cost	46080

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

Alternative 4
Error	0.9
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 5
Error	11.0
Cost	33096

\[\begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -4.5 \cdot 10^{-117}:\\ \;\;\;\;\left|eh + \cos \tan^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot t_1\right|\\ \mathbf{elif}\;ew \leq 2.4 \cdot 10^{-199}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \cdot \left(eh \cdot \cos t\right) + ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]

Alternative 6
Error	11.1
Cost	32968

\[\begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := \tan^{-1} \left(\frac{\frac{eh}{t}}{ew}\right)\\ \mathbf{if}\;ew \leq -9.5 \cdot 10^{-117}:\\ \;\;\;\;\left|eh + \cos t_2 \cdot t_1\right|\\ \mathbf{elif}\;ew \leq 4.4 \cdot 10^{-199}:\\ \;\;\;\;\left|\frac{t}{\frac{\frac{eh}{ew \cdot ew}}{t}} + \left(eh \cdot \cos t\right) \cdot \sin t_2\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]

Alternative 7
Error	1.1
Cost	32832

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

Alternative 8
Error	1.1
Cost	32832

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

Alternative 9
Error	13.4
Cost	27081

\[\begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{t}}{ew}\right)\\ \mathbf{if}\;ew \leq -2.3 \cdot 10^{-117} \lor \neg \left(ew \leq 3.75 \cdot 10^{-202}\right):\\ \;\;\;\;\left|eh + \cos t_1 \cdot \left(ew \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin t_1 + \left(ew \cdot ew\right) \cdot \frac{t \cdot t}{eh}\right|\\ \end{array} \]

Alternative 10
Error	11.2
Cost	27081

\[\begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{t}}{ew}\right)\\ \mathbf{if}\;ew \leq -1.75 \cdot 10^{-116} \lor \neg \left(ew \leq 5.1 \cdot 10^{-199}\right):\\ \;\;\;\;\left|eh + \cos t_1 \cdot \left(ew \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{t}{\frac{\frac{eh}{ew \cdot ew}}{t}} + \left(eh \cdot \cos t\right) \cdot \sin t_1\right|\\ \end{array} \]

Alternative 11
Error	13.7
Cost	26304

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

Alternative 12
Error	42.8
Cost	20288

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

Reproduce

herbie shell --seed 2022340 
(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))))))))