Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.8s

Alternatives: 8

Speedup: N/A×

Specification

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan((-eh / (ew / tan(t))));
	return fabs(((ew * (cos(t) * cos(t_1))) - (eh * (sin(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 * (cos(t) * cos(t_1))) - (eh * (sin(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.cos(t) * Math.cos(t_1))) - (eh * (Math.sin(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.cos(t) * math.cos(t_1))) - (eh * (math.sin(t) * math.sin(t_1)))))

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[((-eh) / N[(ew / N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(ew * N[(N[Cos[t], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. sub0-neg 99.8%

      \[\leadsto \left|\color{blue}{0 - \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. sub-neg 99.8%

      \[\leadsto \left|0 - \color{blue}{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}\right| \]

   4. +-commutative 99.8%

      \[\leadsto \left|0 - \color{blue}{\left(\left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   5. associate--r+ 99.8%

      \[\leadsto \left|\color{blue}{\left(0 - \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. sub0-neg 99.8%

      \[\leadsto \left|\color{blue}{0 - \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. sub-neg 99.8%

      \[\leadsto \left|0 - \color{blue}{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}\right| \]

   4. +-commutative 99.8%

      \[\leadsto \left|0 - \color{blue}{\left(\left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   5. associate--r+ 99.8%

      \[\leadsto \left|\color{blue}{\left(0 - \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

3. Simplified 99.8%

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

   1. expm1-log1p-u 72.3%

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

   2. expm1-udef 54.8%

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

5. Applied egg-rr 56.7%

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

   1. expm1-def 74.2%

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

   2. expm1-log1p 99.8%

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

   3. associate-*r/ 99.8%

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

   4. associate-/l* 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 83.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.7 \cdot 10^{+116}:\\ \;\;\;\;\left|ew - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-t}{\frac{ew}{eh}}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 5.5 \cdot 10^{-130}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -1.7e+116)
   (fabs (- ew (* eh (* (sin t) (sin (atan (/ (- t) (/ ew eh))))))))
   (if (<= eh 5.5e-130)
     (fabs (* ew (cos t)))
     (fabs (- ew (* eh (* (sin t) (sin (atan (/ (- eh) (/ ew (tan t))))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.7e+116) {
		tmp = fabs((ew - (eh * (sin(t) * sin(atan((-t / (ew / eh))))))));
	} else if (eh <= 5.5e-130) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs((ew - (eh * (sin(t) * sin(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 (eh <= (-1.7d+116)) then
        tmp = abs((ew - (eh * (sin(t) * sin(atan((-t / (ew / eh))))))))
    else if (eh <= 5.5d-130) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs((ew - (eh * (sin(t) * sin(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 (eh <= -1.7e+116) {
		tmp = Math.abs((ew - (eh * (Math.sin(t) * Math.sin(Math.atan((-t / (ew / eh))))))));
	} else if (eh <= 5.5e-130) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs((ew - (eh * (Math.sin(t) * Math.sin(Math.atan((-eh / (ew / Math.tan(t)))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -1.7e+116:
		tmp = math.fabs((ew - (eh * (math.sin(t) * math.sin(math.atan((-t / (ew / eh))))))))
	elif eh <= 5.5e-130:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs((ew - (eh * (math.sin(t) * math.sin(math.atan((-eh / (ew / math.tan(t)))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -1.7e+116)
		tmp = abs(Float64(ew - Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(-t) / Float64(ew / eh))))))));
	elseif (eh <= 5.5e-130)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(ew - Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(-eh) / Float64(ew / tan(t)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -1.7e+116)
		tmp = abs((ew - (eh * (sin(t) * sin(atan((-t / (ew / eh))))))));
	elseif (eh <= 5.5e-130)
		tmp = abs((ew * cos(t)));
	else
		tmp = abs((ew - (eh * (sin(t) * sin(atan((-eh / (ew / tan(t)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -1.7e+116], N[Abs[N[(ew - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[((-t) / N[(ew / eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 5.5e-130], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) / N[(ew / N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.70000000000000011e116

   1. Initial program 99.9%

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

      1. fabs-neg 99.9%

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

      2. sub0-neg 99.9%

         \[\leadsto \left|\color{blue}{0 - \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. sub-neg 99.9%

         \[\leadsto \left|0 - \color{blue}{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}\right| \]

      4. +-commutative 99.9%

         \[\leadsto \left|0 - \color{blue}{\left(\left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      5. associate--r+ 99.9%

         \[\leadsto \left|\color{blue}{\left(0 - \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   3. Simplified 99.9%

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

      1. expm1-log1p-u 85.6%

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

      2. expm1-udef 84.9%

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

   5. Applied egg-rr 87.2%

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

      1. expm1-def 87.9%

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

      2. expm1-log1p 99.9%

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

      3. associate-*r/ 99.9%

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

      4. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in t around 0 98.1%

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

   9. Taylor expanded in t around 0 98.1%

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

       1. mul-1-neg 98.1%

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

       2. associate-/l* 98.1%

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

       3. distribute-neg-frac 98.1%

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

   11. Simplified 98.1%

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

   if -1.70000000000000011e116 < eh < 5.50000000000000007e-130

   1. Initial program 99.8%

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

      1. fabs-neg 99.8%

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

      2. sub0-neg 99.8%

         \[\leadsto \left|\color{blue}{0 - \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. sub-neg 99.8%

         \[\leadsto \left|0 - \color{blue}{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}\right| \]

      4. +-commutative 99.8%

         \[\leadsto \left|0 - \color{blue}{\left(\left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      5. associate--r+ 99.8%

         \[\leadsto \left|\color{blue}{\left(0 - \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   3. Simplified 99.8%

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

      1. add-cube-cbrt 98.2%

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

      2. pow3 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. sin-mult 85.8%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right) - \cos \left(t + \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right)}{2}}\right| \]

      2. associate-*r/ 85.8%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right) - \cos \left(t + \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right)\right)}{2}}\right| \]

   7. Applied egg-rr 85.5%

      \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{\frac{eh \cdot \left(\cos \left(t + \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}{2}}\right| \]
   8. Step-by-step derivation

      1. +-inverses 85.5%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. mul0-rgt 85.5%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \frac{\color{blue}{0}}{2}\right| \]

      3. metadata-eval 85.5%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{0}\right| \]

   9. Simplified 85.5%

      \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{0}\right| \]

   10. Taylor expanded in eh around 0 87.1%

       \[\leadsto \left|\color{blue}{\left(\cos t \cdot ew\right) \cdot {1}^{0.3333333333333333}} - 0\right| \]
   11. Step-by-step derivation

       1. pow-base-1 87.1%

          \[\leadsto \left|\left(\cos t \cdot ew\right) \cdot \color{blue}{1} - 0\right| \]

       2. *-rgt-identity 87.1%

          \[\leadsto \left|\color{blue}{\cos t \cdot ew} - 0\right| \]

       3. *-commutative 87.1%

          \[\leadsto \left|\color{blue}{ew \cdot \cos t} - 0\right| \]

   12. Simplified 87.1%

       \[\leadsto \left|\color{blue}{ew \cdot \cos t} - 0\right| \]

   if 5.50000000000000007e-130 < eh

   1. Initial program 99.8%

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

      1. fabs-neg 99.8%

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

      2. sub0-neg 99.8%

         \[\leadsto \left|\color{blue}{0 - \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. sub-neg 99.8%

         \[\leadsto \left|0 - \color{blue}{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}\right| \]

      4. +-commutative 99.8%

         \[\leadsto \left|0 - \color{blue}{\left(\left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      5. associate--r+ 99.8%

         \[\leadsto \left|\color{blue}{\left(0 - \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   3. Simplified 99.8%

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

      1. expm1-log1p-u 75.4%

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

      2. expm1-udef 62.0%

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

   5. Applied egg-rr 66.5%

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

      1. expm1-def 80.0%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r/ 99.8%

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

      4. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 90.8%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.7 \cdot 10^{+116}:\\ \;\;\;\;\left|ew - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-t}{\frac{ew}{eh}}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 5.5 \cdot 10^{-130}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right)\right|\\ \end{array} \]

Alternative 4: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. sub0-neg 99.8%

      \[\leadsto \left|\color{blue}{0 - \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. sub-neg 99.8%

      \[\leadsto \left|0 - \color{blue}{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}\right| \]

   4. +-commutative 99.8%

      \[\leadsto \left|0 - \color{blue}{\left(\left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   5. associate--r+ 99.8%

      \[\leadsto \left|\color{blue}{\left(0 - \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 80.0%

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

   3. associate-*r/ 78.0%

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

   4. associate-/r/ 75.6%

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

   5. *-commutative 75.6%

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

   6. add-sqr-sqrt 34.9%

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

   7. sqrt-unprod 67.6%

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

   8. sqr-neg 67.6%

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

   9. sqrt-unprod 40.1%

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

   10. add-sqr-sqrt 74.8%

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

   11. hypot-1-def 79.0%

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

   12. associate-/r/ 79.0%

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

   13. *-commutative 79.0%

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

5. Applied egg-rr 79.0%

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

6. Taylor expanded in eh around -inf 98.8%

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

   1. mul-1-neg 98.8%

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

   2. distribute-rgt-neg-out 98.8%

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

8. Simplified 98.8%

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

   1. expm1-log1p-u 72.3%

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

   2. expm1-udef 54.8%

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

10. Applied egg-rr 56.8%

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

    1. expm1-def 73.6%

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

    2. expm1-log1p 98.8%

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

    3. associate-*r/ 98.8%

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

12. Simplified 98.8%

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

13. Final simplification 98.8%

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

Alternative 5: 83.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.7 \cdot 10^{+116} \lor \neg \left(eh \leq 9.2 \cdot 10^{-131}\right):\\ \;\;\;\;\left|ew - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-t}{\frac{ew}{eh}}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -1.7e+116) (not (<= eh 9.2e-131)))
   (fabs (- ew (* eh (* (sin t) (sin (atan (/ (- t) (/ ew eh))))))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.7e+116) || !(eh <= 9.2e-131)) {
		tmp = fabs((ew - (eh * (sin(t) * sin(atan((-t / (ew / eh))))))));
	} else {
		tmp = fabs((ew * cos(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 ((eh <= (-1.7d+116)) .or. (.not. (eh <= 9.2d-131))) then
        tmp = abs((ew - (eh * (sin(t) * sin(atan((-t / (ew / eh))))))))
    else
        tmp = abs((ew * cos(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.7e+116) || !(eh <= 9.2e-131)) {
		tmp = Math.abs((ew - (eh * (Math.sin(t) * Math.sin(Math.atan((-t / (ew / eh))))))));
	} else {
		tmp = Math.abs((ew * Math.cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -1.7e+116) or not (eh <= 9.2e-131):
		tmp = math.fabs((ew - (eh * (math.sin(t) * math.sin(math.atan((-t / (ew / eh))))))))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -1.7e+116) || !(eh <= 9.2e-131))
		tmp = abs(Float64(ew - Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(-t) / Float64(ew / eh))))))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -1.7e+116) || ~((eh <= 9.2e-131)))
		tmp = abs((ew - (eh * (sin(t) * sin(atan((-t / (ew / eh))))))));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -1.7e+116], N[Not[LessEqual[eh, 9.2e-131]], $MachinePrecision]], N[Abs[N[(ew - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[((-t) / N[(ew / eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.70000000000000011e116 or 9.20000000000000087e-131 < eh

   1. Initial program 99.8%

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

      1. fabs-neg 99.8%

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

      2. sub0-neg 99.8%

         \[\leadsto \left|\color{blue}{0 - \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. sub-neg 99.8%

         \[\leadsto \left|0 - \color{blue}{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}\right| \]

      4. +-commutative 99.8%

         \[\leadsto \left|0 - \color{blue}{\left(\left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      5. associate--r+ 99.8%

         \[\leadsto \left|\color{blue}{\left(0 - \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   3. Simplified 99.8%

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

      1. expm1-log1p-u 78.8%

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

      2. expm1-udef 69.5%

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

   5. Applied egg-rr 73.4%

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

      1. expm1-def 82.6%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r/ 99.8%

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

      4. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 93.3%

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

   9. Taylor expanded in t around 0 93.3%

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

       1. mul-1-neg 93.3%

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

       2. associate-/l* 93.3%

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

       3. distribute-neg-frac 93.3%

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

   11. Simplified 93.3%

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

   if -1.70000000000000011e116 < eh < 9.20000000000000087e-131

   1. Initial program 99.8%

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

      1. fabs-neg 99.8%

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

      2. sub0-neg 99.8%

         \[\leadsto \left|\color{blue}{0 - \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. sub-neg 99.8%

         \[\leadsto \left|0 - \color{blue}{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}\right| \]

      4. +-commutative 99.8%

         \[\leadsto \left|0 - \color{blue}{\left(\left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      5. associate--r+ 99.8%

         \[\leadsto \left|\color{blue}{\left(0 - \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   3. Simplified 99.8%

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

      1. add-cube-cbrt 98.2%

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

      2. pow3 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. sin-mult 85.8%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right) - \cos \left(t + \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right)}{2}}\right| \]

      2. associate-*r/ 85.8%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right) - \cos \left(t + \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right)\right)}{2}}\right| \]

   7. Applied egg-rr 85.5%

      \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{\frac{eh \cdot \left(\cos \left(t + \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}{2}}\right| \]
   8. Step-by-step derivation

      1. +-inverses 85.5%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. mul0-rgt 85.5%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \frac{\color{blue}{0}}{2}\right| \]

      3. metadata-eval 85.5%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{0}\right| \]

   9. Simplified 85.5%

      \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{0}\right| \]

   10. Taylor expanded in eh around 0 87.1%

       \[\leadsto \left|\color{blue}{\left(\cos t \cdot ew\right) \cdot {1}^{0.3333333333333333}} - 0\right| \]
   11. Step-by-step derivation

       1. pow-base-1 87.1%

          \[\leadsto \left|\left(\cos t \cdot ew\right) \cdot \color{blue}{1} - 0\right| \]

       2. *-rgt-identity 87.1%

          \[\leadsto \left|\color{blue}{\cos t \cdot ew} - 0\right| \]

       3. *-commutative 87.1%

          \[\leadsto \left|\color{blue}{ew \cdot \cos t} - 0\right| \]

   12. Simplified 87.1%

       \[\leadsto \left|\color{blue}{ew \cdot \cos t} - 0\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.7 \cdot 10^{+116} \lor \neg \left(eh \leq 9.2 \cdot 10^{-131}\right):\\ \;\;\;\;\left|ew - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-t}{\frac{ew}{eh}}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]

Alternative 6: 74.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7000000000000 \lor \neg \left(t \leq 1.05 \cdot 10^{-10}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{-t}{\frac{ew}{eh}}\right)\right) - ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -7000000000000.0) (not (<= t 1.05e-10)))
   (fabs (* ew (cos t)))
   (fabs (- (* t (* eh (sin (atan (/ (- t) (/ ew eh)))))) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -7000000000000.0) || !(t <= 1.05e-10)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs(((t * (eh * sin(atan((-t / (ew / eh)))))) - ew));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-7000000000000.0d0)) .or. (.not. (t <= 1.05d-10))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs(((t * (eh * sin(atan((-t / (ew / eh)))))) - ew))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -7000000000000.0) || !(t <= 1.05e-10)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs(((t * (eh * Math.sin(Math.atan((-t / (ew / eh)))))) - ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -7000000000000.0) or not (t <= 1.05e-10):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs(((t * (eh * math.sin(math.atan((-t / (ew / eh)))))) - ew))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -7000000000000.0) || !(t <= 1.05e-10))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(Float64(t * Float64(eh * sin(atan(Float64(Float64(-t) / Float64(ew / eh)))))) - ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -7000000000000.0) || ~((t <= 1.05e-10)))
		tmp = abs((ew * cos(t)));
	else
		tmp = abs(((t * (eh * sin(atan((-t / (ew / eh)))))) - ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -7000000000000.0], N[Not[LessEqual[t, 1.05e-10]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t * N[(eh * N[Sin[N[ArcTan[N[((-t) / N[(ew / eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7e12 or 1.05e-10 < t

   1. Initial program 99.6%

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

      1. fabs-neg 99.6%

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

      2. sub0-neg 99.6%

         \[\leadsto \left|\color{blue}{0 - \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. sub-neg 99.6%

         \[\leadsto \left|0 - \color{blue}{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}\right| \]

      4. +-commutative 99.6%

         \[\leadsto \left|0 - \color{blue}{\left(\left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      5. associate--r+ 99.6%

         \[\leadsto \left|\color{blue}{\left(0 - \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   3. Simplified 99.6%

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

      1. add-cube-cbrt 98.8%

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

      2. pow3 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. sin-mult 57.5%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right) - \cos \left(t + \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right)}{2}}\right| \]

      2. associate-*r/ 57.5%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right) - \cos \left(t + \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right)\right)}{2}}\right| \]

   7. Applied egg-rr 56.0%

      \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{\frac{eh \cdot \left(\cos \left(t + \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}{2}}\right| \]
   8. Step-by-step derivation

      1. +-inverses 56.0%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. mul0-rgt 56.0%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \frac{\color{blue}{0}}{2}\right| \]

      3. metadata-eval 56.0%

         \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{0}\right| \]

   9. Simplified 56.0%

      \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{0}\right| \]

   10. Taylor expanded in eh around 0 57.3%

       \[\leadsto \left|\color{blue}{\left(\cos t \cdot ew\right) \cdot {1}^{0.3333333333333333}} - 0\right| \]
   11. Step-by-step derivation

       1. pow-base-1 57.3%

          \[\leadsto \left|\left(\cos t \cdot ew\right) \cdot \color{blue}{1} - 0\right| \]

       2. *-rgt-identity 57.3%

          \[\leadsto \left|\color{blue}{\cos t \cdot ew} - 0\right| \]

       3. *-commutative 57.3%

          \[\leadsto \left|\color{blue}{ew \cdot \cos t} - 0\right| \]

   12. Simplified 57.3%

       \[\leadsto \left|\color{blue}{ew \cdot \cos t} - 0\right| \]

   if -7e12 < t < 1.05e-10

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. sub0-neg 100.0%

         \[\leadsto \left|\color{blue}{0 - \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. sub-neg 100.0%

         \[\leadsto \left|0 - \color{blue}{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}\right| \]

      4. +-commutative 100.0%

         \[\leadsto \left|0 - \color{blue}{\left(\left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      5. associate--r+ 100.0%

         \[\leadsto \left|\color{blue}{\left(0 - \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   3. Simplified 100.0%

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

      1. expm1-log1p-u 72.0%

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

      2. expm1-udef 51.6%

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

   5. Applied egg-rr 52.4%

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

      1. expm1-def 72.7%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 98.4%

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

   9. Taylor expanded in t around 0 96.6%

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

       1. *-commutative 96.6%

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

       2. associate-*l* 96.6%

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

       3. associate-*r/ 96.6%

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

       4. neg-mul-1 96.6%

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

       5. distribute-rgt-neg-in 96.6%

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

   11. Simplified 96.6%

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

   12. Taylor expanded in t around 0 96.6%

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

       1. mul-1-neg 96.6%

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

       2. associate-/l* 96.6%

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

       3. distribute-neg-frac 96.6%

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

   14. Simplified 96.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7000000000000 \lor \neg \left(t \leq 1.05 \cdot 10^{-10}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{-t}{\frac{ew}{eh}}\right)\right) - ew\right|\\ \end{array} \]

Alternative 7: 62.5% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. sub0-neg 99.8%

      \[\leadsto \left|\color{blue}{0 - \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. sub-neg 99.8%

      \[\leadsto \left|0 - \color{blue}{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}\right| \]

   4. +-commutative 99.8%

      \[\leadsto \left|0 - \color{blue}{\left(\left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   5. associate--r+ 99.8%

      \[\leadsto \left|\color{blue}{\left(0 - \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

3. Simplified 99.8%

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

   1. add-cube-cbrt 98.6%

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

   2. pow3 98.7%

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

5. Applied egg-rr 98.7%

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

   1. sin-mult 66.2%

      \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - eh \cdot \color{blue}{\frac{\cos \left(t - \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right) - \cos \left(t + \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right)}{2}}\right| \]

   2. associate-*r/ 66.2%

      \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{\frac{eh \cdot \left(\cos \left(t - \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right) - \cos \left(t + \tan^{-1} \left(\frac{-eh}{\frac{ew}{\tan t}}\right)\right)\right)}{2}}\right| \]

7. Applied egg-rr 64.1%

   \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{\frac{eh \cdot \left(\cos \left(t + \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}{2}}\right| \]
8. Step-by-step derivation

   1. +-inverses 64.1%

      \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

   2. mul0-rgt 64.1%

      \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \frac{\color{blue}{0}}{2}\right| \]

   3. metadata-eval 64.1%

      \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{0}\right| \]

9. Simplified 64.1%

   \[\leadsto \left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}^{3} - \color{blue}{0}\right| \]

10. Taylor expanded in eh around 0 65.7%

    \[\leadsto \left|\color{blue}{\left(\cos t \cdot ew\right) \cdot {1}^{0.3333333333333333}} - 0\right| \]
11. Step-by-step derivation

    1. pow-base-1 65.7%

       \[\leadsto \left|\left(\cos t \cdot ew\right) \cdot \color{blue}{1} - 0\right| \]

    2. *-rgt-identity 65.7%

       \[\leadsto \left|\color{blue}{\cos t \cdot ew} - 0\right| \]

    3. *-commutative 65.7%

       \[\leadsto \left|\color{blue}{ew \cdot \cos t} - 0\right| \]

12. Simplified 65.7%

    \[\leadsto \left|\color{blue}{ew \cdot \cos t} - 0\right| \]

13. Final simplification 65.7%

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

Alternative 8: 43.1% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(ew)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[ew], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. sub0-neg 99.8%

      \[\leadsto \left|\color{blue}{0 - \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. sub-neg 99.8%

      \[\leadsto \left|0 - \color{blue}{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right)}\right| \]

   4. +-commutative 99.8%

      \[\leadsto \left|0 - \color{blue}{\left(\left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   5. associate--r+ 99.8%

      \[\leadsto \left|\color{blue}{\left(0 - \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

3. Simplified 99.8%

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

   1. expm1-log1p-u 72.3%

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

   2. expm1-udef 54.8%

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

5. Applied egg-rr 56.7%

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

   1. expm1-def 74.2%

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

   2. expm1-log1p 99.8%

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

   3. associate-*r/ 99.8%

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

   4. associate-/l* 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 78.8%

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

9. Taylor expanded in t around 0 58.2%

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

    1. *-commutative 58.2%

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

    2. associate-*l* 58.2%

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

    3. associate-*r/ 58.2%

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

    4. neg-mul-1 58.2%

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

    5. distribute-rgt-neg-in 58.2%

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

11. Simplified 58.2%

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

12. Taylor expanded in ew around inf 45.7%

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

13. Final simplification 45.7%

    \[\leadsto \left|ew\right| \]

Reproduce

herbie shell --seed 2023222 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))