Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 21.4s

Alternatives: 11

Speedup: N/A×

Specification

Math

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

FPCore

(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) {
	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
    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) {
	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):
	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)
	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)
	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_] := 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]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(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) {
	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
    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) {
	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):
	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)
	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)
	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_] := 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]]

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (+
   (* (/ 1.0 (hypot 1.0 (/ (/ eh (tan t)) ew))) (* ew (sin t)))
   (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))

C

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

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((1.0 / Math.hypot(1.0, ((eh / Math.tan(t)) / ew))) * (ew * Math.sin(t))) + ((eh * Math.cos(t)) * Math.sin(Math.atan(((eh / ew) / Math.tan(t)))))));
}

Python

def code(eh, ew, t):
	return math.fabs((((1.0 / math.hypot(1.0, ((eh / math.tan(t)) / ew))) * (ew * math.sin(t))) + ((eh * math.cos(t)) * math.sin(math.atan(((eh / ew) / math.tan(t)))))))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((1.0 / hypot(1.0, ((eh / tan(t)) / ew))) * (ew * sin(t))) + ((eh * cos(t)) * sin(atan(((eh / ew) / tan(t)))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(ew * N[Sin[t], $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]

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. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/r* 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-/l/ 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (+
   (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t)))))
   (* (cos (atan (/ (/ eh t) ew))) (* ew (sin t))))))

C

double code(double eh, double ew, double t) {
	return fabs((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + (cos(atan(((eh / t) / ew))) * (ew * sin(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((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + (cos(atan(((eh / t) / ew))) * (ew * sin(t)))))
end function

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / ew) / tan(t))))) + Float64(cos(atan(Float64(Float64(eh / t) / ew))) * Float64(ew * sin(t)))))
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(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] + N[(N[Cos[N[ArcTan[N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(ew * N[Sin[t], $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. Taylor expanded in t around 0 99.5%

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

   1. associate-/r* 88.8%

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

4. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 98.4% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (+ (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))) (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	return fabs((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + (ew * sin(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((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + (ew * sin(t))))
end function

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / ew) / tan(t))))) + Float64(ew * sin(t))))
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(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] + N[(ew * N[Sin[t], $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. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/r* 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-/l/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in eh around 0 99.3%

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

7. Final simplification 99.3%

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

Alternative 4: 95.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ t_2 := ew \cdot \sin t\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{+118} \lor \neg \left(t \leq 1.25 \cdot 10^{+169}\right):\\ \;\;\;\;\left|t_2 + t_1 \cdot \sin \tan^{-1} \left(-0.3333333333333333 \cdot \frac{t \cdot eh}{ew} + \frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_2 + t_1 \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{t}}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))) (t_2 (* ew (sin t))))
   (if (or (<= t -2.4e+118) (not (<= t 1.25e+169)))
     (fabs
      (+
       t_2
       (*
        t_1
        (sin
         (atan (+ (* -0.3333333333333333 (/ (* t eh) ew)) (/ eh (* ew t))))))))
     (fabs (+ t_2 (* t_1 (sin (atan (/ (/ eh t) ew)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double t_2 = ew * sin(t);
	double tmp;
	if ((t <= -2.4e+118) || !(t <= 1.25e+169)) {
		tmp = fabs((t_2 + (t_1 * sin(atan(((-0.3333333333333333 * ((t * eh) / ew)) + (eh / (ew * t))))))));
	} else {
		tmp = fabs((t_2 + (t_1 * sin(atan(((eh / t) / ew))))));
	}
	return tmp;
}

Fortran

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
    real(8) :: t_2
    real(8) :: tmp
    t_1 = eh * cos(t)
    t_2 = ew * sin(t)
    if ((t <= (-2.4d+118)) .or. (.not. (t <= 1.25d+169))) then
        tmp = abs((t_2 + (t_1 * sin(atan((((-0.3333333333333333d0) * ((t * eh) / ew)) + (eh / (ew * t))))))))
    else
        tmp = abs((t_2 + (t_1 * sin(atan(((eh / t) / ew))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.cos(t);
	double t_2 = ew * Math.sin(t);
	double tmp;
	if ((t <= -2.4e+118) || !(t <= 1.25e+169)) {
		tmp = Math.abs((t_2 + (t_1 * Math.sin(Math.atan(((-0.3333333333333333 * ((t * eh) / ew)) + (eh / (ew * t))))))));
	} else {
		tmp = Math.abs((t_2 + (t_1 * Math.sin(Math.atan(((eh / t) / ew))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.cos(t)
	t_2 = ew * math.sin(t)
	tmp = 0
	if (t <= -2.4e+118) or not (t <= 1.25e+169):
		tmp = math.fabs((t_2 + (t_1 * math.sin(math.atan(((-0.3333333333333333 * ((t * eh) / ew)) + (eh / (ew * t))))))))
	else:
		tmp = math.fabs((t_2 + (t_1 * math.sin(math.atan(((eh / t) / ew))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	t_2 = Float64(ew * sin(t))
	tmp = 0.0
	if ((t <= -2.4e+118) || !(t <= 1.25e+169))
		tmp = abs(Float64(t_2 + Float64(t_1 * sin(atan(Float64(Float64(-0.3333333333333333 * Float64(Float64(t * eh) / ew)) + Float64(eh / Float64(ew * t))))))));
	else
		tmp = abs(Float64(t_2 + Float64(t_1 * sin(atan(Float64(Float64(eh / t) / ew))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * cos(t);
	t_2 = ew * sin(t);
	tmp = 0.0;
	if ((t <= -2.4e+118) || ~((t <= 1.25e+169)))
		tmp = abs((t_2 + (t_1 * sin(atan(((-0.3333333333333333 * ((t * eh) / ew)) + (eh / (ew * t))))))));
	else
		tmp = abs((t_2 + (t_1 * sin(atan(((eh / t) / ew))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -2.4e+118], N[Not[LessEqual[t, 1.25e+169]], $MachinePrecision]], N[Abs[N[(t$95$2 + N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(-0.3333333333333333 * N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] + N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$2 + N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.4e118 or 1.25000000000000004e169 < t

   1. Initial program 99.6%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. cos-atan 99.6%

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

      2. hypot-1-def 99.6%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. associate-/l/ 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in eh around 0 99.5%

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

   7. Taylor expanded in t around 0 95.3%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(-0.3333333333333333 \cdot \frac{t \cdot eh}{ew} + \frac{eh}{t \cdot ew}\right)}\right| \]

   if -2.4e118 < t < 1.25000000000000004e169

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. cos-atan 99.9%

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

      2. hypot-1-def 99.9%

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

      3. associate-/r* 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-/l/ 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in eh around 0 99.3%

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

   7. Taylor expanded in t around 0 98.7%

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

      1. associate-/r* 98.7%

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

   9. Simplified 98.7%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{t}}{ew}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+118} \lor \neg \left(t \leq 1.25 \cdot 10^{+169}\right):\\ \;\;\;\;\left|ew \cdot \sin t + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(-0.3333333333333333 \cdot \frac{t \cdot eh}{ew} + \frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{t}}{ew}\right)\right|\\ \end{array} \]

Alternative 5: 89.1% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (+ (* ew (sin t)) (* (* eh (cos t)) (sin (atan (/ (/ eh t) ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs(((ew * sin(t)) + ((eh * cos(t)) * sin(atan(((eh / t) / ew))))));
}

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)) + ((eh * cos(t)) * sin(atan(((eh / t) / ew))))))
end function

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * sin(t)) + Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / t) / ew))))))
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / t), $MachinePrecision] / ew), $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. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/r* 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-/l/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in eh around 0 99.3%

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

7. Taylor expanded in t around 0 88.8%

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

   1. associate-/r* 88.8%

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

9. Simplified 88.8%

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

10. Final simplification 88.8%

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

Alternative 6: 79.2% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (+ (* ew (sin t)) (* eh (sin (atan (/ eh (* ew (tan t)))))))))

C

double code(double eh, double ew, double t) {
	return fabs(((ew * sin(t)) + (eh * sin(atan((eh / (ew * tan(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)) + (eh * sin(atan((eh / (ew * tan(t))))))))
end function

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * sin(t)) + Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t))))))))
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $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. Step-by-step derivation

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/r* 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-/l/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in eh around 0 99.3%

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

7. Taylor expanded in t around 0 77.7%

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

8. Final simplification 77.7%

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

Alternative 7: 78.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(ew, \frac{\sin t}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}, eh\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (fma ew (/ (sin t) (hypot 1.0 (/ (/ eh ew) (tan t)))) eh)))

C

double code(double eh, double ew, double t) {
	return fabs(fma(ew, (sin(t) / hypot(1.0, ((eh / ew) / tan(t)))), eh));
}

Julia

function code(eh, ew, t)
	return abs(fma(ew, Float64(sin(t) / hypot(1.0, Float64(Float64(eh / ew) / tan(t)))), eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[Sin[t], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + eh), $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. Step-by-step derivation

   1. associate-*l* 99.8%

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

   2. fma-def 99.8%

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

   3. associate-/l/ 99.8%

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

   4. associate-/r* 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*l* 99.8%

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

   7. associate-/l/ 99.8%

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

   8. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. sin-atan 56.5%

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

   4. associate-*r/ 52.2%

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

   5. associate-/l/ 51.2%

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

   6. hypot-1-def 59.1%

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

   7. associate-/l/ 59.4%

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

5. Applied egg-rr 59.4%

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

6. Taylor expanded in t around 0 77.0%

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

   1. add-cube-cbrt 76.2%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\left(\sqrt[3]{\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)} \cdot \sqrt[3]{\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right) \cdot \sqrt[3]{\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}, eh\right)\right| \]

   2. pow3 76.2%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{{\left(\sqrt[3]{\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right)}^{3}}, eh\right)\right| \]

   3. associate-/l/ 76.2%

      \[\leadsto \left|\mathsf{fma}\left(ew, {\left(\sqrt[3]{\sin t \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot \tan t}\right)}}\right)}^{3}, eh\right)\right| \]

   4. associate-/r* 76.2%

      \[\leadsto \left|\mathsf{fma}\left(ew, {\left(\sqrt[3]{\sin t \cdot \cos \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}}\right)}^{3}, eh\right)\right| \]

8. Applied egg-rr 76.2%

   \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{{\left(\sqrt[3]{\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right)}^{3}}, eh\right)\right| \]
9. Step-by-step derivation

   1. rem-cube-cbrt 77.0%

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

   2. expm1-log1p-u 76.9%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)}, eh\right)\right| \]

   3. expm1-udef 61.6%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{e^{\mathsf{log1p}\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)} - 1}, eh\right)\right| \]

   4. cos-atan 61.6%

      \[\leadsto \left|\mathsf{fma}\left(ew, e^{\mathsf{log1p}\left(\sin t \cdot \color{blue}{\frac{1}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right)} - 1, eh\right)\right| \]

   5. un-div-inv 61.6%

      \[\leadsto \left|\mathsf{fma}\left(ew, e^{\mathsf{log1p}\left(\color{blue}{\frac{\sin t}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right)} - 1, eh\right)\right| \]

   6. hypot-1-def 61.6%

      \[\leadsto \left|\mathsf{fma}\left(ew, e^{\mathsf{log1p}\left(\frac{\sin t}{\color{blue}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}}\right)} - 1, eh\right)\right| \]

10. Applied egg-rr 61.6%

    \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin t}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}\right)} - 1}, eh\right)\right| \]
11. Step-by-step derivation

    1. expm1-def 76.9%

       \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin t}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}\right)\right)}, eh\right)\right| \]

    2. expm1-log1p 77.0%

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

12. Simplified 77.0%

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

13. Final simplification 77.0%

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

Alternative 8: 78.9% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (fma ew (* (sin t) (cos (atan (/ (/ eh t) ew)))) eh)))

C

double code(double eh, double ew, double t) {
	return fabs(fma(ew, (sin(t) * cos(atan(((eh / t) / ew)))), eh));
}

Julia

function code(eh, ew, t)
	return abs(fma(ew, Float64(sin(t) * cos(atan(Float64(Float64(eh / t) / ew)))), eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[Sin[t], $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + eh), $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. Step-by-step derivation

   1. associate-*l* 99.8%

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

   2. fma-def 99.8%

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

   3. associate-/l/ 99.8%

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

   4. associate-/r* 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*l* 99.8%

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

   7. associate-/l/ 99.8%

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

   8. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. sin-atan 56.5%

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

   4. associate-*r/ 52.2%

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

   5. associate-/l/ 51.2%

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

   6. hypot-1-def 59.1%

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

   7. associate-/l/ 59.4%

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

5. Applied egg-rr 59.4%

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

6. Taylor expanded in t around 0 77.0%

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

7. Taylor expanded in t around 0 76.9%

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

8. Final simplification 76.9%

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

Alternative 9: 64.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + \left(1 + -0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t -1.35e+154)
   (fabs eh)
   (fabs
    (+
     (* ew (sin t))
     (* (+ 1.0 (* -0.5 (* t t))) (* eh (sin (atan (/ eh (* ew t))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -1.35e+154) {
		tmp = fabs(eh);
	} else {
		tmp = fabs(((ew * sin(t)) + ((1.0 + (-0.5 * (t * t))) * (eh * sin(atan((eh / (ew * t))))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.35d+154)) then
        tmp = abs(eh)
    else
        tmp = abs(((ew * sin(t)) + ((1.0d0 + ((-0.5d0) * (t * t))) * (eh * sin(atan((eh / (ew * t))))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -1.35e+154) {
		tmp = Math.abs(eh);
	} else {
		tmp = Math.abs(((ew * Math.sin(t)) + ((1.0 + (-0.5 * (t * t))) * (eh * Math.sin(Math.atan((eh / (ew * t))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if t <= -1.35e+154:
		tmp = math.fabs(eh)
	else:
		tmp = math.fabs(((ew * math.sin(t)) + ((1.0 + (-0.5 * (t * t))) * (eh * math.sin(math.atan((eh / (ew * t))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= -1.35e+154)
		tmp = abs(eh);
	else
		tmp = abs(Float64(Float64(ew * sin(t)) + Float64(Float64(1.0 + Float64(-0.5 * Float64(t * t))) * Float64(eh * sin(atan(Float64(eh / Float64(ew * t))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (t <= -1.35e+154)
		tmp = abs(eh);
	else
		tmp = abs(((ew * sin(t)) + ((1.0 + (-0.5 * (t * t))) * (eh * sin(atan((eh / (ew * t))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, -1.35e+154], N[Abs[eh], $MachinePrecision], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(-0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35000000000000003e154

   1. Initial program 99.6%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. associate-*l* 99.6%

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

      2. fma-def 99.6%

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

      3. associate-/l/ 99.6%

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

      4. associate-/r* 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*l* 99.6%

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

      7. associate-/l/ 99.6%

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

      8. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r* 99.6%

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

      2. *-commutative 99.6%

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

      3. sin-atan 66.2%

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

      4. associate-*r/ 57.4%

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

      5. associate-/l/ 57.4%

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

      6. hypot-1-def 62.5%

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

      7. associate-/l/ 62.6%

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

   5. Applied egg-rr 62.6%

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

   6. Taylor expanded in t around 0 51.6%

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

   7. Taylor expanded in t around 0 11.0%

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

      1. associate-/l/ 11.0%

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

   9. Simplified 11.0%

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

   10. Taylor expanded in ew around 0 15.2%

       \[\leadsto \left|\color{blue}{eh}\right| \]

   if -1.35000000000000003e154 < t

   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. Step-by-step derivation

      1. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. associate-/r* 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-/l/ 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in eh around 0 99.3%

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

   7. Taylor expanded in t around 0 74.2%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \color{blue}{\left(\sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right) \cdot eh + -0.5 \cdot \left({t}^{2} \cdot \left(\sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right) \cdot eh\right)\right)\right)}\right| \]
   8. Step-by-step derivation

      1. associate-*r* 74.2%

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \left(\sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right) \cdot eh + \color{blue}{\left(-0.5 \cdot {t}^{2}\right) \cdot \left(\sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right) \cdot eh\right)}\right)\right| \]

      2. distribute-rgt1-in 74.2%

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \color{blue}{\left(-0.5 \cdot {t}^{2} + 1\right) \cdot \left(\sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right) \cdot eh\right)}\right| \]

      3. unpow2 74.2%

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \left(-0.5 \cdot \color{blue}{\left(t \cdot t\right)} + 1\right) \cdot \left(\sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right) \cdot eh\right)\right| \]

      4. *-commutative 74.2%

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \left(-0.5 \cdot \left(t \cdot t\right) + 1\right) \cdot \color{blue}{\left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)}\right| \]

      5. associate-/l/ 74.2%

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \left(-0.5 \cdot \left(t \cdot t\right) + 1\right) \cdot \left(eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right)\right| \]

   9. Simplified 74.2%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \color{blue}{\left(-0.5 \cdot \left(t \cdot t\right) + 1\right) \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)}\right| \]

   10. Taylor expanded in t around 0 75.2%

       \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \left(-0.5 \cdot \left(t \cdot t\right) + 1\right) \cdot \left(eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{t \cdot ew}\right)}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + \left(1 + -0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \end{array} \]

Alternative 10: 55.7% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (+ eh (* ew (/ t (hypot 1.0 (/ (/ eh ew) (tan t))))))))

C

double code(double eh, double ew, double t) {
	return fabs((eh + (ew * (t / hypot(1.0, ((eh / ew) / tan(t)))))));
}

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((eh + (ew * (t / Math.hypot(1.0, ((eh / ew) / Math.tan(t)))))));
}

Python

def code(eh, ew, t):
	return math.fabs((eh + (ew * (t / math.hypot(1.0, ((eh / ew) / math.tan(t)))))))

Julia

function code(eh, ew, t)
	return abs(Float64(eh + Float64(ew * Float64(t / hypot(1.0, Float64(Float64(eh / ew) / tan(t)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((eh + (ew * (t / hypot(1.0, ((eh / ew) / tan(t)))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(eh + N[(ew * N[(t / N[Sqrt[1.0 ^ 2 + N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision] ^ 2], $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. Step-by-step derivation

   1. associate-*l* 99.8%

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

   2. fma-def 99.8%

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

   3. associate-/l/ 99.8%

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

   4. associate-/r* 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*l* 99.8%

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

   7. associate-/l/ 99.8%

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

   8. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. sin-atan 56.5%

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

   4. associate-*r/ 52.2%

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

   5. associate-/l/ 51.2%

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

   6. hypot-1-def 59.1%

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

   7. associate-/l/ 59.4%

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

5. Applied egg-rr 59.4%

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

6. Taylor expanded in t around 0 77.0%

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

7. Taylor expanded in t around 0 56.0%

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

   1. associate-/l/ 56.0%

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

9. Simplified 56.0%

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

    1. fma-udef 56.0%

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

    2. cos-atan 56.5%

       \[\leadsto \left|ew \cdot \left(t \cdot \color{blue}{\frac{1}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right) + eh\right| \]

    3. un-div-inv 56.5%

       \[\leadsto \left|ew \cdot \color{blue}{\frac{t}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}} + eh\right| \]

    4. hypot-1-def 56.5%

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

11. Applied egg-rr 56.5%

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

12. Final simplification 56.5%

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

Alternative 11: 42.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \left|eh\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs eh))

C

double code(double eh, double ew, double t) {
	return fabs(eh);
}

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(eh)
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(eh);
}

Python

def code(eh, ew, t):
	return math.fabs(eh)

Julia

function code(eh, ew, t)
	return abs(eh)
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(eh);
end

Wolfram

code[eh_, ew_, t_] := N[Abs[eh], $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. Step-by-step derivation

   1. associate-*l* 99.8%

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

   2. fma-def 99.8%

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

   3. associate-/l/ 99.8%

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

   4. associate-/r* 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*l* 99.8%

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

   7. associate-/l/ 99.8%

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

   8. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. sin-atan 56.5%

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

   4. associate-*r/ 52.2%

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

   5. associate-/l/ 51.2%

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

   6. hypot-1-def 59.1%

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

   7. associate-/l/ 59.4%

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

5. Applied egg-rr 59.4%

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

6. Taylor expanded in t around 0 77.0%

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

7. Taylor expanded in t around 0 56.0%

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

   1. associate-/l/ 56.0%

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

9. Simplified 56.0%

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

10. Taylor expanded in ew around 0 41.4%

    \[\leadsto \left|\color{blue}{eh}\right| \]

11. Final simplification 41.4%

    \[\leadsto \left|eh\right| \]

Reproduce

herbie shell --seed 2023175 
(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))))))))