Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 23.2s

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

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) (tan t))))
   (fabs
    (+
     (* (* ew (sin t)) (/ 1.0 (hypot 1.0 t_1)))
     (* (sin (atan t_1)) (* eh (cos t)))))))

C

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

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / Math.tan(t);
	return Math.abs((((ew * Math.sin(t)) * (1.0 / Math.hypot(1.0, t_1))) + (Math.sin(Math.atan(t_1)) * (eh * Math.cos(t)))));
}

Python

def code(eh, ew, t):
	t_1 = (eh / ew) / math.tan(t)
	return math.fabs((((ew * math.sin(t)) * (1.0 / math.hypot(1.0, t_1))) + (math.sin(math.atan(t_1)) * (eh * math.cos(t)))))

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / tan(t))
	return abs(Float64(Float64(Float64(ew * sin(t)) * Float64(1.0 / hypot(1.0, t_1))) + Float64(sin(atan(t_1)) * Float64(eh * cos(t)))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = (eh / ew) / tan(t);
	tmp = abs((((ew * sin(t)) * (1.0 / hypot(1.0, t_1))) + (sin(atan(t_1)) * (eh * cos(t)))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision] * N[(eh * N[Cos[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. Step-by-step derivation

   1. associate-/l/ 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

   4. associate-/l/ 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/l/ 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

   4. associate-/l/ 99.8%

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

3. Applied egg-rr 99.8%

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

   1. un-div-inv 99.8%

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

   2. associate-/l* 99.7%

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

   3. associate-/r* 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[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. Taylor expanded in t around 0 98.9%

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

   1. associate-/r* 98.9%

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

4. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 4: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + N[(eh * (-N[Cos[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. fma-def 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 62.2%

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

   3. associate-*r/ 59.3%

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

   4. associate-/l/ 59.5%

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

   5. hypot-1-def 68.0%

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

   6. associate-/l/ 72.0%

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

5. Applied egg-rr 72.0%

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

   1. associate-/l* 83.0%

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

   2. associate-/r/ 79.0%

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

   3. associate-/l/ 78.9%

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

   4. associate-/l/ 79.1%

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

7. Simplified 79.1%

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

8. Taylor expanded in eh around -inf 98.8%

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

   1. mul-1-neg 98.8%

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

   2. distribute-rgt-neg-out 98.8%

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

10. Simplified 98.8%

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

    1. associate-/l/ 79.1%

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

    2. cos-atan 79.1%

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

    3. metadata-eval 79.1%

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

    4. hypot-udef 79.1%

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

    5. inv-pow 79.1%

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

    6. sqr-pow 79.1%

       \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{{\left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)}^{\left(\frac{-1}{2}\right)}}, \frac{eh \cdot \cos t}{\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)} \cdot \frac{eh}{\tan t \cdot ew}\right)\right| \]

    7. associate-/r* 79.1%

       \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, {\left(\mathsf{hypot}\left(1, \color{blue}{\frac{eh}{ew \cdot \tan t}}\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)}^{\left(\frac{-1}{2}\right)}, \frac{eh \cdot \cos t}{\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)} \cdot \frac{eh}{\tan t \cdot ew}\right)\right| \]

    8. metadata-eval 79.1%

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

    9. associate-/r* 79.1%

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

    10. metadata-eval 79.1%

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

12. Applied egg-rr 98.8%

    \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{{\left(\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)}^{-0.5}}, \cos t \cdot \left(-eh\right)\right)\right| \]
13. Step-by-step derivation

    1. pow-sqr 79.1%

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

    2. metadata-eval 79.1%

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

    3. unpow-1 79.1%

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

14. Simplified 98.8%

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

15. Final simplification 98.8%

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

Alternative 5: 86.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.6 \cdot 10^{-20} \lor \neg \left(ew \leq 6.1 \cdot 10^{-130}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right), -eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.6e-20) (not (<= ew 6.1e-130)))
   (fabs (fma (* ew (sin t)) (cos (atan (/ eh (* ew (tan t))))) (- eh)))
   (fabs (* eh (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.6e-20) || !(ew <= 6.1e-130)) {
		tmp = fabs(fma((ew * sin(t)), cos(atan((eh / (ew * tan(t))))), -eh));
	} else {
		tmp = fabs((eh * cos(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.6e-20) || !(ew <= 6.1e-130))
		tmp = abs(fma(Float64(ew * sin(t)), cos(atan(Float64(eh / Float64(ew * tan(t))))), Float64(-eh)));
	else
		tmp = abs(Float64(eh * cos(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.6e-20], N[Not[LessEqual[ew, 6.1e-130]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + (-eh)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.59999999999999985e-20 or 6.09999999999999996e-130 < ew

   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. fma-def 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*r* 99.8%

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

      2. sin-atan 79.3%

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

      3. associate-*r/ 74.9%

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

      4. associate-/l/ 74.8%

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

      5. hypot-1-def 77.7%

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

      6. associate-/l/ 78.6%

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

   5. Applied egg-rr 78.6%

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

      1. associate-/l* 91.8%

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

      2. associate-/r/ 90.9%

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

      3. associate-/l/ 90.8%

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

      4. associate-/l/ 91.1%

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

   7. Simplified 91.1%

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

   8. Taylor expanded in eh around -inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. distribute-rgt-neg-out 98.9%

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

   10. Simplified 98.9%

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

   11. Taylor expanded in t around 0 84.4%

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

       1. mul-1-neg 84.4%

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

   13. Simplified 84.4%

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

   if -1.59999999999999985e-20 < ew < 6.09999999999999996e-130

   1. Initial program 99.7%

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

      1. fma-def 99.7%

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

      2. associate-/l/ 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r* 99.7%

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

      2. sin-atan 32.8%

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

      3. associate-*r/ 32.6%

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

      4. associate-/l/ 33.0%

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

      5. hypot-1-def 51.2%

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

      6. associate-/l/ 60.7%

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

   5. Applied egg-rr 60.7%

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

      1. associate-/l* 67.9%

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

      2. associate-/r/ 58.4%

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

      3. associate-/l/ 58.3%

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

      4. associate-/l/ 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in eh around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. distribute-rgt-neg-out 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in t around 0 72.8%

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

   12. Taylor expanded in ew around 0 91.4%

       \[\leadsto \left|\color{blue}{-1 \cdot \left(\cos t \cdot eh\right)}\right| \]
   13. Step-by-step derivation

       1. mul-1-neg 91.4%

          \[\leadsto \left|\color{blue}{-\cos t \cdot eh}\right| \]

       2. distribute-rgt-neg-out 91.4%

          \[\leadsto \left|\color{blue}{\cos t \cdot \left(-eh\right)}\right| \]

   14. Simplified 91.4%

       \[\leadsto \left|\color{blue}{\cos t \cdot \left(-eh\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.6 \cdot 10^{-20} \lor \neg \left(ew \leq 6.1 \cdot 10^{-130}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right), -eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \]

Alternative 6: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + N[(eh * N[Cos[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. fma-def 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 62.2%

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

   3. associate-*r/ 59.3%

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

   4. associate-/l/ 59.5%

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

   5. hypot-1-def 68.0%

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

   6. associate-/l/ 72.0%

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

5. Applied egg-rr 72.0%

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

   1. associate-/l* 83.0%

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

   2. associate-/r/ 79.0%

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

   3. associate-/l/ 78.9%

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

   4. associate-/l/ 79.1%

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

7. Simplified 79.1%

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

   1. associate-/l/ 79.1%

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

   2. cos-atan 79.1%

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

   3. metadata-eval 79.1%

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

   4. hypot-udef 79.1%

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

   5. inv-pow 79.1%

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

   6. sqr-pow 79.1%

      \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{{\left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)}^{\left(\frac{-1}{2}\right)}}, \frac{eh \cdot \cos t}{\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)} \cdot \frac{eh}{\tan t \cdot ew}\right)\right| \]

   7. associate-/r* 79.1%

      \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, {\left(\mathsf{hypot}\left(1, \color{blue}{\frac{eh}{ew \cdot \tan t}}\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)}^{\left(\frac{-1}{2}\right)}, \frac{eh \cdot \cos t}{\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)} \cdot \frac{eh}{\tan t \cdot ew}\right)\right| \]

   8. metadata-eval 79.1%

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

   9. associate-/r* 79.1%

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

   10. metadata-eval 79.1%

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

9. Applied egg-rr 79.1%

   \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{{\left(\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)}^{-0.5}}, \frac{eh \cdot \cos t}{\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)} \cdot \frac{eh}{\tan t \cdot ew}\right)\right| \]
10. Step-by-step derivation

    1. pow-sqr 79.1%

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

    2. metadata-eval 79.1%

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

    3. unpow-1 79.1%

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

11. Simplified 79.1%

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

12. Taylor expanded in eh around inf 98.7%

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

13. Final simplification 98.7%

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

Alternative 7: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/l/ 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

   4. associate-/l/ 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in ew around inf 98.5%

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

5. Final simplification 98.5%

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

Alternative 8: 86.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -7.2 \cdot 10^{-20} \lor \neg \left(ew \leq 4.2 \cdot 10^{-129}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}, eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -7.2e-20) (not (<= ew 4.2e-129)))
   (fabs (fma (* ew (sin t)) (/ 1.0 (hypot 1.0 (/ eh (* ew (tan t))))) eh))
   (fabs (* eh (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -7.2e-20) || !(ew <= 4.2e-129)) {
		tmp = fabs(fma((ew * sin(t)), (1.0 / hypot(1.0, (eh / (ew * tan(t))))), eh));
	} else {
		tmp = fabs((eh * cos(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -7.2e-20) || !(ew <= 4.2e-129))
		tmp = abs(fma(Float64(ew * sin(t)), Float64(1.0 / hypot(1.0, Float64(eh / Float64(ew * tan(t))))), eh));
	else
		tmp = abs(Float64(eh * cos(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -7.2e-20], N[Not[LessEqual[ew, 4.2e-129]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -7.19999999999999948e-20 or 4.2e-129 < ew

   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. fma-def 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*r* 99.8%

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

      2. sin-atan 79.3%

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

      3. associate-*r/ 74.9%

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

      4. associate-/l/ 74.8%

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

      5. hypot-1-def 77.7%

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

      6. associate-/l/ 78.6%

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

   5. Applied egg-rr 78.6%

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

      1. associate-/l* 91.8%

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

      2. associate-/r/ 90.9%

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

      3. associate-/l/ 90.8%

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

      4. associate-/l/ 91.1%

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

   7. Simplified 91.1%

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

      1. associate-/l/ 91.1%

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

      2. cos-atan 91.1%

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

      3. metadata-eval 91.1%

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

      4. hypot-udef 91.1%

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

      5. inv-pow 91.1%

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

      6. sqr-pow 91.1%

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{{\left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)}^{\left(\frac{-1}{2}\right)}}, \frac{eh \cdot \cos t}{\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)} \cdot \frac{eh}{\tan t \cdot ew}\right)\right| \]

      7. associate-/r* 91.1%

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, {\left(\mathsf{hypot}\left(1, \color{blue}{\frac{eh}{ew \cdot \tan t}}\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)}^{\left(\frac{-1}{2}\right)}, \frac{eh \cdot \cos t}{\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)} \cdot \frac{eh}{\tan t \cdot ew}\right)\right| \]

      8. metadata-eval 91.1%

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

      9. associate-/r* 91.1%

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

      10. metadata-eval 91.1%

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

   9. Applied egg-rr 91.1%

      \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{{\left(\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)}^{-0.5}}, \frac{eh \cdot \cos t}{\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)} \cdot \frac{eh}{\tan t \cdot ew}\right)\right| \]
   10. Step-by-step derivation

       1. pow-sqr 91.1%

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

       2. metadata-eval 91.1%

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

       3. unpow-1 91.1%

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

   11. Simplified 91.1%

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

   12. Taylor expanded in t around 0 84.4%

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

   if -7.19999999999999948e-20 < ew < 4.2e-129

   1. Initial program 99.7%

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

      1. fma-def 99.7%

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

      2. associate-/l/ 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r* 99.7%

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

      2. sin-atan 32.8%

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

      3. associate-*r/ 32.6%

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

      4. associate-/l/ 33.0%

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

      5. hypot-1-def 51.2%

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

      6. associate-/l/ 60.7%

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

   5. Applied egg-rr 60.7%

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

      1. associate-/l* 67.9%

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

      2. associate-/r/ 58.4%

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

      3. associate-/l/ 58.3%

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

      4. associate-/l/ 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in eh around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. distribute-rgt-neg-out 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in t around 0 72.8%

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

   12. Taylor expanded in ew around 0 91.4%

       \[\leadsto \left|\color{blue}{-1 \cdot \left(\cos t \cdot eh\right)}\right| \]
   13. Step-by-step derivation

       1. mul-1-neg 91.4%

          \[\leadsto \left|\color{blue}{-\cos t \cdot eh}\right| \]

       2. distribute-rgt-neg-out 91.4%

          \[\leadsto \left|\color{blue}{\cos t \cdot \left(-eh\right)}\right| \]

   14. Simplified 91.4%

       \[\leadsto \left|\color{blue}{\cos t \cdot \left(-eh\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -7.2 \cdot 10^{-20} \lor \neg \left(ew \leq 4.2 \cdot 10^{-129}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}, eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \]

Alternative 9: 62.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.26 \cdot 10^{-73} \lor \neg \left(t \leq 4.2 \cdot 10^{-5}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t 1.26e-73) (not (<= t 4.2e-5)))
   (fabs (* eh (cos t)))
   (fabs (* (* ew t) (cos (atan (/ (/ eh ew) (tan t))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= 1.26e-73) || !(t <= 4.2e-5)) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = fabs(((ew * t) * cos(atan(((eh / ew) / tan(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 <= 1.26d-73) .or. (.not. (t <= 4.2d-5))) then
        tmp = abs((eh * cos(t)))
    else
        tmp = abs(((ew * t) * cos(atan(((eh / ew) / tan(t))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= 1.26e-73) || !(t <= 4.2e-5)) {
		tmp = Math.abs((eh * Math.cos(t)));
	} else {
		tmp = Math.abs(((ew * t) * Math.cos(Math.atan(((eh / ew) / Math.tan(t))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= 1.26e-73) or not (t <= 4.2e-5):
		tmp = math.fabs((eh * math.cos(t)))
	else:
		tmp = math.fabs(((ew * t) * math.cos(math.atan(((eh / ew) / math.tan(t))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= 1.26e-73) || !(t <= 4.2e-5))
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = abs(Float64(Float64(ew * t) * cos(atan(Float64(Float64(eh / ew) / tan(t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= 1.26e-73) || ~((t <= 4.2e-5)))
		tmp = abs((eh * cos(t)));
	else
		tmp = abs(((ew * t) * cos(atan(((eh / ew) / tan(t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, 1.26e-73], N[Not[LessEqual[t, 4.2e-5]], $MachinePrecision]], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * t), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.26000000000000001e-73 or 4.19999999999999977e-5 < 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. fma-def 99.7%

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

      2. associate-/l/ 99.7%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*r* 99.7%

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

      2. sin-atan 60.9%

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

      3. associate-*r/ 57.7%

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

      4. associate-/l/ 57.9%

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

      5. hypot-1-def 66.6%

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

      6. associate-/l/ 71.0%

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

   5. Applied egg-rr 71.0%

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

      1. associate-/l* 82.1%

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

      2. associate-/r/ 77.7%

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

      3. associate-/l/ 77.6%

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

      4. associate-/l/ 77.8%

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

   7. Simplified 77.8%

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

   8. Taylor expanded in eh around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-out 98.7%

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

   10. Simplified 98.7%

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

   11. Taylor expanded in t around 0 60.4%

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

   12. Taylor expanded in ew around 0 66.9%

       \[\leadsto \left|\color{blue}{-1 \cdot \left(\cos t \cdot eh\right)}\right| \]
   13. Step-by-step derivation

       1. mul-1-neg 66.9%

          \[\leadsto \left|\color{blue}{-\cos t \cdot eh}\right| \]

       2. distribute-rgt-neg-out 66.9%

          \[\leadsto \left|\color{blue}{\cos t \cdot \left(-eh\right)}\right| \]

   14. Simplified 66.9%

       \[\leadsto \left|\color{blue}{\cos t \cdot \left(-eh\right)}\right| \]

   if 1.26000000000000001e-73 < t < 4.19999999999999977e-5

   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. fma-def 99.9%

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

      2. associate-/l/ 99.9%

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

      3. associate-*l* 99.9%

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

      4. associate-/l/ 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r* 99.9%

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

      2. sin-atan 79.5%

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

      3. associate-*r/ 79.1%

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

      4. associate-/l/ 79.1%

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

      5. hypot-1-def 84.6%

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

      6. associate-/l/ 84.7%

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

   5. Applied egg-rr 84.7%

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

      1. associate-/l* 94.7%

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

      2. associate-/r/ 94.7%

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

      3. associate-/l/ 94.6%

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

      4. associate-/l/ 94.7%

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

   7. Simplified 94.7%

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

   8. Taylor expanded in eh around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. distribute-rgt-neg-out 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in t around 0 98.4%

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

   12. Taylor expanded in t around inf 78.0%

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

       1. *-commutative 78.0%

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

       2. associate-*r* 78.0%

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

       3. associate-/l/ 78.0%

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

   14. Simplified 78.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.26 \cdot 10^{-73} \lor \neg \left(t \leq 4.2 \cdot 10^{-5}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \end{array} \]

Alternative 10: 62.7% accurate, 4.5× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((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((eh * cos(t)))
end function

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(eh * cos(t)))
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(eh * N[Cos[t], $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. fma-def 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 62.2%

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

   3. associate-*r/ 59.3%

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

   4. associate-/l/ 59.5%

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

   5. hypot-1-def 68.0%

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

   6. associate-/l/ 72.0%

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

5. Applied egg-rr 72.0%

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

   1. associate-/l* 83.0%

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

   2. associate-/r/ 79.0%

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

   3. associate-/l/ 78.9%

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

   4. associate-/l/ 79.1%

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

7. Simplified 79.1%

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

8. Taylor expanded in eh around -inf 98.8%

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

   1. mul-1-neg 98.8%

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

   2. distribute-rgt-neg-out 98.8%

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

10. Simplified 98.8%

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

11. Taylor expanded in t around 0 63.2%

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

12. Taylor expanded in ew around 0 63.7%

    \[\leadsto \left|\color{blue}{-1 \cdot \left(\cos t \cdot eh\right)}\right| \]
13. Step-by-step derivation

    1. mul-1-neg 63.7%

       \[\leadsto \left|\color{blue}{-\cos t \cdot eh}\right| \]

    2. distribute-rgt-neg-out 63.7%

       \[\leadsto \left|\color{blue}{\cos t \cdot \left(-eh\right)}\right| \]

14. Simplified 63.7%

    \[\leadsto \left|\color{blue}{\cos t \cdot \left(-eh\right)}\right| \]

15. Final simplification 63.7%

    \[\leadsto \left|eh \cdot \cos t\right| \]

Alternative 11: 43.4% 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. fma-def 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 62.2%

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

   3. associate-*r/ 59.3%

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

   4. associate-/l/ 59.5%

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

   5. hypot-1-def 68.0%

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

   6. associate-/l/ 72.0%

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

5. Applied egg-rr 72.0%

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

   1. associate-/l* 83.0%

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

   2. associate-/r/ 79.0%

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

   3. associate-/l/ 78.9%

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

   4. associate-/l/ 79.1%

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

7. Simplified 79.1%

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

8. Taylor expanded in eh around -inf 98.8%

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

   1. mul-1-neg 98.8%

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

   2. distribute-rgt-neg-out 98.8%

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

10. Simplified 98.8%

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

11. Taylor expanded in t around 0 63.2%

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

12. Taylor expanded in t around 0 39.9%

    \[\leadsto \left|\color{blue}{-1 \cdot eh}\right| \]
13. Step-by-step derivation

    1. neg-mul-1 39.9%

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

14. Simplified 39.9%

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

15. Final simplification 39.9%

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

Reproduce

herbie shell --seed 2023214 
(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))))))))