Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 24.8s

Alternatives: 14

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, 1.1× speedup

Math

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

FPCore

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

C

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

Python

def code(eh, ew, t):
	return math.fabs(((ew * (math.cos(t) * (1.0 / math.hypot(1.0, (math.tan(t) * (eh / ew)))))) - (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(cos(t) * Float64(1.0 / hypot(1.0, Float64(tan(t) * Float64(eh / ew)))))) - 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 * (cos(t) * (1.0 / hypot(1.0, (tan(t) * (eh / ew)))))) - (eh * (sin(t) * sin(atan((-eh / (ew / tan(t)))))))));
end

Wolfram

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

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 48.7%

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

   5. sqrt-unprod 95.5%

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

   6. sqr-neg 95.5%

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

   7. sqrt-unprod 51.1%

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

   8. add-sqr-sqrt 99.8%

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

   9. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 91.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{-t}{\frac{ew}{eh}}\right)\\ \mathbf{if}\;ew \leq -1.55 \cdot 10^{-105}:\\ \;\;\;\;\left|ew \cdot \left(\cos t \cdot \frac{1}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right) - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \left(t \cdot 0.3333333333333333 + \frac{-1}{t}\right)}\right)\right)\right|\\ \mathbf{elif}\;ew \leq 5.1 \cdot 10^{-100}:\\ \;\;\;\;\left|ew \cdot \left(\cos t \cdot \cos t_1\right) - eh \cdot \left(\sin t \cdot \sin t_1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(ew \cdot t\right) + \frac{ew}{t}}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (- t) (/ ew eh)))))
   (if (<= ew -1.55e-105)
     (fabs
      (-
       (* ew (* (cos t) (/ 1.0 (hypot 1.0 (* (tan t) (/ eh ew))))))
       (*
        eh
        (*
         (sin t)
         (sin (atan (/ eh (* ew (+ (* t 0.3333333333333333) (/ -1.0 t))))))))))
     (if (<= ew 5.1e-100)
       (fabs (- (* ew (* (cos t) (cos t_1))) (* eh (* (sin t) (sin t_1)))))
       (fabs
        (-
         (* ew (cos t))
         (*
          eh
          (*
           (sin t)
           (sin
            (atan
             (/ (- eh) (+ (* -0.3333333333333333 (* ew t)) (/ ew t)))))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((-t / (ew / eh)));
	double tmp;
	if (ew <= -1.55e-105) {
		tmp = fabs(((ew * (cos(t) * (1.0 / hypot(1.0, (tan(t) * (eh / ew)))))) - (eh * (sin(t) * sin(atan((eh / (ew * ((t * 0.3333333333333333) + (-1.0 / t))))))))));
	} else if (ew <= 5.1e-100) {
		tmp = fabs(((ew * (cos(t) * cos(t_1))) - (eh * (sin(t) * sin(t_1)))));
	} else {
		tmp = fabs(((ew * cos(t)) - (eh * (sin(t) * sin(atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((-t / (ew / eh)));
	double tmp;
	if (ew <= -1.55e-105) {
		tmp = Math.abs(((ew * (Math.cos(t) * (1.0 / Math.hypot(1.0, (Math.tan(t) * (eh / ew)))))) - (eh * (Math.sin(t) * Math.sin(Math.atan((eh / (ew * ((t * 0.3333333333333333) + (-1.0 / t))))))))));
	} else if (ew <= 5.1e-100) {
		tmp = Math.abs(((ew * (Math.cos(t) * Math.cos(t_1))) - (eh * (Math.sin(t) * Math.sin(t_1)))));
	} else {
		tmp = Math.abs(((ew * Math.cos(t)) - (eh * (Math.sin(t) * Math.sin(Math.atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((-t / (ew / eh)))
	tmp = 0
	if ew <= -1.55e-105:
		tmp = math.fabs(((ew * (math.cos(t) * (1.0 / math.hypot(1.0, (math.tan(t) * (eh / ew)))))) - (eh * (math.sin(t) * math.sin(math.atan((eh / (ew * ((t * 0.3333333333333333) + (-1.0 / t))))))))))
	elif ew <= 5.1e-100:
		tmp = math.fabs(((ew * (math.cos(t) * math.cos(t_1))) - (eh * (math.sin(t) * math.sin(t_1)))))
	else:
		tmp = math.fabs(((ew * math.cos(t)) - (eh * (math.sin(t) * math.sin(math.atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(-t) / Float64(ew / eh)))
	tmp = 0.0
	if (ew <= -1.55e-105)
		tmp = abs(Float64(Float64(ew * Float64(cos(t) * Float64(1.0 / hypot(1.0, Float64(tan(t) * Float64(eh / ew)))))) - Float64(eh * Float64(sin(t) * sin(atan(Float64(eh / Float64(ew * Float64(Float64(t * 0.3333333333333333) + Float64(-1.0 / t))))))))));
	elseif (ew <= 5.1e-100)
		tmp = abs(Float64(Float64(ew * Float64(cos(t) * cos(t_1))) - Float64(eh * Float64(sin(t) * sin(t_1)))));
	else
		tmp = abs(Float64(Float64(ew * cos(t)) - Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(-eh) / Float64(Float64(-0.3333333333333333 * Float64(ew * t)) + Float64(ew / t)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((-t / (ew / eh)));
	tmp = 0.0;
	if (ew <= -1.55e-105)
		tmp = abs(((ew * (cos(t) * (1.0 / hypot(1.0, (tan(t) * (eh / ew)))))) - (eh * (sin(t) * sin(atan((eh / (ew * ((t * 0.3333333333333333) + (-1.0 / t))))))))));
	elseif (ew <= 5.1e-100)
		tmp = abs(((ew * (cos(t) * cos(t_1))) - (eh * (sin(t) * sin(t_1)))));
	else
		tmp = abs(((ew * cos(t)) - (eh * (sin(t) * sin(atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[((-t) / N[(ew / eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.55e-105], N[Abs[N[(N[(ew * N[(N[Cos[t], $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[(N[(t * 0.3333333333333333), $MachinePrecision] + N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 5.1e-100], 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], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) / N[(N[(-0.3333333333333333 * N[(ew * t), $MachinePrecision]), $MachinePrecision] + N[(ew / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.55000000000000007e-105

   1. Initial program 99.7%

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

      1. fabs-neg 99.7%

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

      2. sub0-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. associate--r+ 99.7%

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

   3. Simplified 99.7%

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

      1. cos-atan 99.7%

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

      2. hypot-1-def 99.7%

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

      3. div-inv 99.7%

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

      4. add-sqr-sqrt 48.0%

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

      5. sqrt-unprod 97.3%

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

      6. sqr-neg 97.3%

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

      7. sqrt-unprod 51.7%

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

      8. add-sqr-sqrt 99.7%

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

      9. clear-num 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in t around 0 94.6%

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

   9. Taylor expanded in ew around -inf 94.6%

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

   if -1.55000000000000007e-105 < ew < 5.0999999999999999e-100

   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. Taylor expanded in t around 0 98.9%

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

      1. mul-1-neg 29.0%

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

      2. associate-/l* 29.0%

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

      3. distribute-neg-frac 29.0%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in t around 0 98.9%

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

      1. mul-1-neg 29.0%

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

      2. associate-/l* 29.0%

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

      3. distribute-neg-frac 29.0%

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

   9. Simplified 98.9%

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

   if 5.0999999999999999e-100 < ew

   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. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. div-inv 99.8%

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

      4. add-sqr-sqrt 48.6%

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

      5. sqrt-unprod 89.1%

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

      6. sqr-neg 89.1%

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

      7. sqrt-unprod 51.2%

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

      8. add-sqr-sqrt 99.8%

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

      9. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 92.3%

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

   9. Taylor expanded in t around 0 92.4%

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

   10. Taylor expanded in ew around 0 92.4%

       \[\leadsto \left|\color{blue}{\cos t \cdot ew} - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(t \cdot ew\right) + \frac{ew}{t}}\right)\right)\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.55 \cdot 10^{-105}:\\ \;\;\;\;\left|ew \cdot \left(\cos t \cdot \frac{1}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right) - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \left(t \cdot 0.3333333333333333 + \frac{-1}{t}\right)}\right)\right)\right|\\ \mathbf{elif}\;ew \leq 5.1 \cdot 10^{-100}:\\ \;\;\;\;\left|ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{-t}{\frac{ew}{eh}}\right)\right) - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-t}{\frac{ew}{eh}}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(ew \cdot t\right) + \frac{ew}{t}}\right)\right)\right|\\ \end{array} \]

Alternative 3: 91.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{-t}{\frac{ew}{eh}}\right)\\ \mathbf{if}\;ew \leq -1.55 \cdot 10^{-105} \lor \neg \left(ew \leq 1.9 \cdot 10^{-99}\right):\\ \;\;\;\;\left|ew \cdot \cos t - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(ew \cdot t\right) + \frac{ew}{t}}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t \cdot \cos t_1\right) - eh \cdot \left(\sin t \cdot \sin t_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (- t) (/ ew eh)))))
   (if (or (<= ew -1.55e-105) (not (<= ew 1.9e-99)))
     (fabs
      (-
       (* ew (cos t))
       (*
        eh
        (*
         (sin t)
         (sin
          (atan (/ (- eh) (+ (* -0.3333333333333333 (* ew t)) (/ ew 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((-t / (ew / eh)));
	double tmp;
	if ((ew <= -1.55e-105) || !(ew <= 1.9e-99)) {
		tmp = fabs(((ew * cos(t)) - (eh * (sin(t) * sin(atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))));
	} else {
		tmp = fabs(((ew * (cos(t) * cos(t_1))) - (eh * (sin(t) * sin(t_1)))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan((-t / (ew / eh)))
    if ((ew <= (-1.55d-105)) .or. (.not. (ew <= 1.9d-99))) then
        tmp = abs(((ew * cos(t)) - (eh * (sin(t) * sin(atan((-eh / (((-0.3333333333333333d0) * (ew * t)) + (ew / t)))))))))
    else
        tmp = abs(((ew * (cos(t) * cos(t_1))) - (eh * (sin(t) * sin(t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((-t / (ew / eh)));
	double tmp;
	if ((ew <= -1.55e-105) || !(ew <= 1.9e-99)) {
		tmp = Math.abs(((ew * Math.cos(t)) - (eh * (Math.sin(t) * Math.sin(Math.atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))));
	} else {
		tmp = Math.abs(((ew * (Math.cos(t) * Math.cos(t_1))) - (eh * (Math.sin(t) * Math.sin(t_1)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((-t / (ew / eh)))
	tmp = 0
	if (ew <= -1.55e-105) or not (ew <= 1.9e-99):
		tmp = math.fabs(((ew * math.cos(t)) - (eh * (math.sin(t) * math.sin(math.atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))))
	else:
		tmp = math.fabs(((ew * (math.cos(t) * math.cos(t_1))) - (eh * (math.sin(t) * math.sin(t_1)))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(-t) / Float64(ew / eh)))
	tmp = 0.0
	if ((ew <= -1.55e-105) || !(ew <= 1.9e-99))
		tmp = abs(Float64(Float64(ew * cos(t)) - Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(-eh) / Float64(Float64(-0.3333333333333333 * Float64(ew * t)) + Float64(ew / t)))))))));
	else
		tmp = abs(Float64(Float64(ew * Float64(cos(t) * cos(t_1))) - Float64(eh * Float64(sin(t) * sin(t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((-t / (ew / eh)));
	tmp = 0.0;
	if ((ew <= -1.55e-105) || ~((ew <= 1.9e-99)))
		tmp = abs(((ew * cos(t)) - (eh * (sin(t) * sin(atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))));
	else
		tmp = abs(((ew * (cos(t) * cos(t_1))) - (eh * (sin(t) * sin(t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[((-t) / N[(ew / eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[ew, -1.55e-105], N[Not[LessEqual[ew, 1.9e-99]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) / N[(N[(-0.3333333333333333 * N[(ew * t), $MachinePrecision]), $MachinePrecision] + N[(ew / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $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. Split input into 2 regimes

2. if ew < -1.55000000000000007e-105 or 1.8999999999999998e-99 < ew

   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. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. div-inv 99.8%

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

      4. add-sqr-sqrt 48.3%

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

      5. sqrt-unprod 93.2%

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

      6. sqr-neg 93.2%

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

      7. sqrt-unprod 51.5%

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

      8. add-sqr-sqrt 99.8%

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

      9. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 93.5%

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

   9. Taylor expanded in t around 0 93.1%

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

   10. Taylor expanded in ew around 0 93.1%

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

   if -1.55000000000000007e-105 < ew < 1.8999999999999998e-99

   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. Taylor expanded in t around 0 98.9%

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

      1. mul-1-neg 29.0%

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

      2. associate-/l* 29.0%

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

      3. distribute-neg-frac 29.0%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in t around 0 98.9%

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

      1. mul-1-neg 29.0%

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

      2. associate-/l* 29.0%

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

      3. distribute-neg-frac 29.0%

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

   9. Simplified 98.9%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.55 \cdot 10^{-105} \lor \neg \left(ew \leq 1.9 \cdot 10^{-99}\right):\\ \;\;\;\;\left|ew \cdot \cos t - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(ew \cdot t\right) + \frac{ew}{t}}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{-t}{\frac{ew}{eh}}\right)\right) - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-t}{\frac{ew}{eh}}\right)\right)\right|\\ \end{array} \]

Alternative 4: 89.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \cos t - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(ew \cdot t\right) + \frac{ew}{t}}\right)\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* ew (cos t))
   (*
    eh
    (*
     (sin t)
     (sin (atan (/ (- eh) (+ (* -0.3333333333333333 (* ew t)) (/ ew t))))))))))

C

double code(double eh, double ew, double t) {
	return fabs(((ew * cos(t)) - (eh * (sin(t) * sin(atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / 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)) - (eh * (sin(t) * sin(atan((-eh / (((-0.3333333333333333d0) * (ew * t)) + (ew / t)))))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((ew * Math.cos(t)) - (eh * (Math.sin(t) * Math.sin(Math.atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))));
}

Python

def code(eh, ew, t):
	return math.fabs(((ew * math.cos(t)) - (eh * (math.sin(t) * math.sin(math.atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * cos(t)) - Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(-eh) / Float64(Float64(-0.3333333333333333 * Float64(ew * t)) + Float64(ew / t)))))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(t)) - (eh * (sin(t) * sin(atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) / N[(N[(-0.3333333333333333 * N[(ew * t), $MachinePrecision]), $MachinePrecision] + N[(ew / 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. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 48.7%

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

   5. sqrt-unprod 95.5%

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

   6. sqr-neg 95.5%

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

   7. sqrt-unprod 51.1%

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

   8. add-sqr-sqrt 99.8%

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

   9. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 89.4%

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

9. Taylor expanded in t around 0 88.7%

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

10. Taylor expanded in ew around 0 88.7%

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

11. Final simplification 88.7%

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

Alternative 5: 89.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \cos t - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(3 \cdot \frac{\frac{eh}{t}}{ew}\right)\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (- (* ew (cos t)) (* eh (* (sin t) (sin (atan (* 3.0 (/ (/ eh t) ew)))))))))

C

double code(double eh, double ew, double t) {
	return fabs(((ew * cos(t)) - (eh * (sin(t) * sin(atan((3.0 * ((eh / t) / ew))))))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(((ew * cos(t)) - (eh * (sin(t) * sin(atan((3.0d0 * ((eh / t) / ew))))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((ew * Math.cos(t)) - (eh * (Math.sin(t) * Math.sin(Math.atan((3.0 * ((eh / t) / ew))))))));
}

Python

def code(eh, ew, t):
	return math.fabs(((ew * math.cos(t)) - (eh * (math.sin(t) * math.sin(math.atan((3.0 * ((eh / t) / ew))))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * cos(t)) - Float64(eh * Float64(sin(t) * sin(atan(Float64(3.0 * Float64(Float64(eh / t) / ew))))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(t)) - (eh * (sin(t) * sin(atan((3.0 * ((eh / t) / ew))))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(3.0 * N[(N[(eh / t), $MachinePrecision] / ew), $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. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 48.7%

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

   5. sqrt-unprod 95.5%

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

   6. sqr-neg 95.5%

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

   7. sqrt-unprod 51.1%

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

   8. add-sqr-sqrt 99.8%

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

   9. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 89.4%

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

9. Taylor expanded in t around 0 88.7%

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

10. Taylor expanded in t around inf 88.6%

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

    1. associate-/r* 88.6%

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

12. Simplified 88.6%

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

13. Final simplification 88.6%

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

Alternative 6: 89.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \cos t - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh}{ew} \cdot \frac{3}{t}\right)\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (- (* ew (cos t)) (* eh (* (sin t) (sin (atan (* (/ eh ew) (/ 3.0 t)))))))))

C

double code(double eh, double ew, double t) {
	return fabs(((ew * cos(t)) - (eh * (sin(t) * sin(atan(((eh / ew) * (3.0 / 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)) - (eh * (sin(t) * sin(atan(((eh / ew) * (3.0d0 / t))))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((ew * Math.cos(t)) - (eh * (Math.sin(t) * Math.sin(Math.atan(((eh / ew) * (3.0 / t))))))));
}

Python

def code(eh, ew, t):
	return math.fabs(((ew * math.cos(t)) - (eh * (math.sin(t) * math.sin(math.atan(((eh / ew) * (3.0 / t))))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * cos(t)) - Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(eh / ew) * Float64(3.0 / t))))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(t)) - (eh * (sin(t) * sin(atan(((eh / ew) * (3.0 / t))))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[(3.0 / t), $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. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 48.7%

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

   5. sqrt-unprod 95.5%

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

   6. sqr-neg 95.5%

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

   7. sqrt-unprod 51.1%

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

   8. add-sqr-sqrt 99.8%

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

   9. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 89.4%

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

9. Taylor expanded in t around 0 88.7%

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

10. Taylor expanded in t around inf 88.6%

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

    1. associate-*r/ 88.6%

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

    2. times-frac 88.6%

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

12. Simplified 88.6%

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

13. Final simplification 88.6%

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

Alternative 7: 79.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -5 \cdot 10^{+121} \lor \neg \left(ew \leq 1.5 \cdot 10^{-50}\right):\\ \;\;\;\;\left|ew \cdot \left(\cos t \cdot \frac{1}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(ew \cdot t\right) + \frac{ew}{t}}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -5e+121) (not (<= ew 1.5e-50)))
   (fabs (* ew (* (cos t) (/ 1.0 (hypot 1.0 (* (tan t) (/ eh ew)))))))
   (fabs
    (-
     ew
     (*
      eh
      (*
       (sin t)
       (sin
        (atan (/ (- eh) (+ (* -0.3333333333333333 (* ew t)) (/ ew t)))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -5e+121) || !(ew <= 1.5e-50)) {
		tmp = fabs((ew * (cos(t) * (1.0 / hypot(1.0, (tan(t) * (eh / ew)))))));
	} else {
		tmp = fabs((ew - (eh * (sin(t) * sin(atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -5e+121) || !(ew <= 1.5e-50)) {
		tmp = Math.abs((ew * (Math.cos(t) * (1.0 / Math.hypot(1.0, (Math.tan(t) * (eh / ew)))))));
	} else {
		tmp = Math.abs((ew - (eh * (Math.sin(t) * Math.sin(Math.atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -5e+121) or not (ew <= 1.5e-50):
		tmp = math.fabs((ew * (math.cos(t) * (1.0 / math.hypot(1.0, (math.tan(t) * (eh / ew)))))))
	else:
		tmp = math.fabs((ew - (eh * (math.sin(t) * math.sin(math.atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -5e+121) || !(ew <= 1.5e-50))
		tmp = abs(Float64(ew * Float64(cos(t) * Float64(1.0 / hypot(1.0, Float64(tan(t) * Float64(eh / ew)))))));
	else
		tmp = abs(Float64(ew - Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(-eh) / Float64(Float64(-0.3333333333333333 * Float64(ew * t)) + Float64(ew / t)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -5e+121) || ~((ew <= 1.5e-50)))
		tmp = abs((ew * (cos(t) * (1.0 / hypot(1.0, (tan(t) * (eh / ew)))))));
	else
		tmp = abs((ew - (eh * (sin(t) * sin(atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -5e+121], N[Not[LessEqual[ew, 1.5e-50]], $MachinePrecision]], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) / N[(N[(-0.3333333333333333 * N[(ew * t), $MachinePrecision]), $MachinePrecision] + N[(ew / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.00000000000000007e121 or 1.49999999999999995e-50 < ew

   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. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. div-inv 99.8%

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

      4. add-sqr-sqrt 47.9%

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

      5. sqrt-unprod 90.4%

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

      6. sqr-neg 90.4%

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

      7. sqrt-unprod 51.9%

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

      8. add-sqr-sqrt 99.8%

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

      9. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

   7. Simplified 99.8%

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

      1. sin-mult 87.2%

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

      2. associate-*r/ 87.2%

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

   9. Applied egg-rr 86.6%

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

       1. +-inverses 86.8%

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

       2. *-commutative 86.8%

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

       3. associate-/l* 86.8%

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

       4. div0 86.8%

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

   11. Simplified 86.6%

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

   if -5.00000000000000007e121 < ew < 1.49999999999999995e-50

   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. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. div-inv 99.8%

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

      4. add-sqr-sqrt 49.2%

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

      5. sqrt-unprod 98.9%

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

      6. sqr-neg 98.9%

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

      7. sqrt-unprod 50.6%

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

      8. add-sqr-sqrt 99.8%

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

      9. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 87.1%

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

   9. Taylor expanded in t around 0 85.9%

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

   10. Taylor expanded in t around 0 79.6%

       \[\leadsto \left|\color{blue}{ew} - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(t \cdot ew\right) + \frac{ew}{t}}\right)\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5 \cdot 10^{+121} \lor \neg \left(ew \leq 1.5 \cdot 10^{-50}\right):\\ \;\;\;\;\left|ew \cdot \left(\cos t \cdot \frac{1}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(ew \cdot t\right) + \frac{ew}{t}}\right)\right)\right|\\ \end{array} \]

Alternative 8: 79.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(ew \cdot t\right) + \frac{ew}{t}}\right)\\ \mathbf{if}\;ew \leq -5.4 \cdot 10^{+119} \lor \neg \left(ew \leq 1.55 \cdot 10^{-50}\right):\\ \;\;\;\;\left|ew \cdot \left(\cos t \cdot \cos t_1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - eh \cdot \left(\sin t \cdot \sin t_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (- eh) (+ (* -0.3333333333333333 (* ew t)) (/ ew t))))))
   (if (or (<= ew -5.4e+119) (not (<= ew 1.55e-50)))
     (fabs (* ew (* (cos t) (cos t_1))))
     (fabs (- ew (* eh (* (sin t) (sin t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t))));
	double tmp;
	if ((ew <= -5.4e+119) || !(ew <= 1.55e-50)) {
		tmp = fabs((ew * (cos(t) * cos(t_1))));
	} else {
		tmp = fabs((ew - (eh * (sin(t) * sin(t_1)))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan((-eh / (((-0.3333333333333333d0) * (ew * t)) + (ew / t))))
    if ((ew <= (-5.4d+119)) .or. (.not. (ew <= 1.55d-50))) then
        tmp = abs((ew * (cos(t) * cos(t_1))))
    else
        tmp = abs((ew - (eh * (sin(t) * sin(t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t))));
	double tmp;
	if ((ew <= -5.4e+119) || !(ew <= 1.55e-50)) {
		tmp = Math.abs((ew * (Math.cos(t) * Math.cos(t_1))));
	} else {
		tmp = Math.abs((ew - (eh * (Math.sin(t) * Math.sin(t_1)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t))))
	tmp = 0
	if (ew <= -5.4e+119) or not (ew <= 1.55e-50):
		tmp = math.fabs((ew * (math.cos(t) * math.cos(t_1))))
	else:
		tmp = math.fabs((ew - (eh * (math.sin(t) * math.sin(t_1)))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(-eh) / Float64(Float64(-0.3333333333333333 * Float64(ew * t)) + Float64(ew / t))))
	tmp = 0.0
	if ((ew <= -5.4e+119) || !(ew <= 1.55e-50))
		tmp = abs(Float64(ew * Float64(cos(t) * cos(t_1))));
	else
		tmp = abs(Float64(ew - Float64(eh * Float64(sin(t) * sin(t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((-eh / ((-0.3333333333333333 * (ew * t)) + (ew / t))));
	tmp = 0.0;
	if ((ew <= -5.4e+119) || ~((ew <= 1.55e-50)))
		tmp = abs((ew * (cos(t) * cos(t_1))));
	else
		tmp = abs((ew - (eh * (sin(t) * sin(t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[((-eh) / N[(N[(-0.3333333333333333 * N[(ew * t), $MachinePrecision]), $MachinePrecision] + N[(ew / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[ew, -5.4e+119], N[Not[LessEqual[ew, 1.55e-50]], $MachinePrecision]], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.3999999999999997e119 or 1.5500000000000001e-50 < ew

   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. sin-mult 87.2%

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

      2. associate-*r/ 87.2%

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

   5. Applied egg-rr 86.8%

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

      1. +-inverses 86.8%

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

      2. *-commutative 86.8%

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

      3. associate-/l* 86.8%

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

      4. div0 86.8%

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

   7. Simplified 86.8%

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

   8. Taylor expanded in t around 0 86.8%

      \[\leadsto \left|ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{-eh}{\color{blue}{-0.3333333333333333 \cdot \left(t \cdot ew\right) + \frac{ew}{t}}}\right)\right) - 0\right| \]

   if -5.3999999999999997e119 < ew < 1.5500000000000001e-50

   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. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. div-inv 99.8%

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

      4. add-sqr-sqrt 49.2%

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

      5. sqrt-unprod 98.9%

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

      6. sqr-neg 98.9%

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

      7. sqrt-unprod 50.6%

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

      8. add-sqr-sqrt 99.8%

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

      9. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 87.1%

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

   9. Taylor expanded in t around 0 85.9%

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

   10. Taylor expanded in t around 0 79.6%

       \[\leadsto \left|\color{blue}{ew} - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(t \cdot ew\right) + \frac{ew}{t}}\right)\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.4 \cdot 10^{+119} \lor \neg \left(ew \leq 1.55 \cdot 10^{-50}\right):\\ \;\;\;\;\left|ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(ew \cdot t\right) + \frac{ew}{t}}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{-eh}{-0.3333333333333333 \cdot \left(ew \cdot t\right) + \frac{ew}{t}}\right)\right)\right|\\ \end{array} \]

Alternative 9: 61.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. hypot-1-def 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 48.7%

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

   5. sqrt-unprod 95.5%

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

   6. sqr-neg 95.5%

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

   7. sqrt-unprod 51.1%

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

   8. add-sqr-sqrt 99.8%

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

   9. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-*r/ 99.8%

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

7. Simplified 99.8%

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

   1. sin-mult 62.6%

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

   2. associate-*r/ 62.6%

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

9. Applied egg-rr 60.1%

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

    1. +-inverses 60.4%

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

    2. *-commutative 60.4%

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

    3. associate-/l* 60.4%

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

    4. div0 60.4%

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

11. Simplified 60.1%

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

12. Final simplification 60.1%

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

Alternative 10: 52.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] * N[Cos[N[ArcTan[N[((-t) / N[(ew / eh), $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. sin-mult 62.6%

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

   2. associate-*r/ 62.6%

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

5. Applied egg-rr 60.4%

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

   1. +-inverses 60.4%

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

   2. *-commutative 60.4%

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

   3. associate-/l* 60.4%

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

   4. div0 60.4%

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

7. Simplified 60.4%

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

8. Taylor expanded in t around 0 50.1%

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

   1. mul-1-neg 50.1%

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

   2. associate-/l* 50.1%

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

   3. distribute-neg-frac 50.1%

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

10. Simplified 50.1%

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

11. Final simplification 50.1%

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

Alternative 11: 42.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / ew), $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. sin-mult 62.6%

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

   2. associate-*r/ 62.6%

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

5. Applied egg-rr 60.4%

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

   1. +-inverses 60.4%

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

   2. *-commutative 60.4%

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

   3. associate-/l* 60.4%

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

   4. div0 60.4%

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

7. Simplified 60.4%

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

   1. add-sqr-sqrt 28.7%

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

   2. pow2 28.7%

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

9. Applied egg-rr 28.7%

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

10. Taylor expanded in t around 0 41.4%

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

11. Final simplification 41.4%

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

Alternative 12: 42.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $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. sin-mult 62.6%

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

   2. associate-*r/ 62.6%

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

5. Applied egg-rr 60.4%

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

   1. +-inverses 60.4%

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

   2. *-commutative 60.4%

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

   3. associate-/l* 60.4%

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

   4. div0 60.4%

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

7. Simplified 60.4%

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

8. Taylor expanded in t around 0 41.4%

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

   1. *-commutative 41.4%

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

   2. mul-1-neg 41.4%

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

   3. associate-*r/ 41.4%

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

   4. distribute-rgt-neg-in 41.4%

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

   5. distribute-neg-frac 41.4%

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

10. Simplified 41.4%

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

    1. cos-atan 41.0%

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

12. Applied egg-rr 41.0%

    \[\leadsto \left|ew \cdot \color{blue}{\frac{1}{\sqrt{1 + \left(\tan t \cdot \frac{-eh}{ew}\right) \cdot \left(\tan t \cdot \frac{-eh}{ew}\right)}}} - 0\right| \]
13. Step-by-step derivation

    1. hypot-1-def 41.1%

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

    2. *-commutative 41.1%

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

    3. associate-/r/ 41.1%

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

    4. hypot-1-def 41.1%

       \[\leadsto \left|ew \cdot \frac{1}{\color{blue}{\sqrt{1 + \frac{-eh}{\frac{ew}{\tan t}} \cdot \frac{-eh}{\frac{ew}{\tan t}}}}} - 0\right| \]

    5. distribute-frac-neg 41.1%

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

    6. neg-mul-1 41.1%

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

    7. associate-/l* 41.1%

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

    8. *-commutative 41.1%

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

    9. distribute-frac-neg 41.1%

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

    10. associate-/r/ 41.0%

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

    11. *-commutative 41.0%

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

    12. associate-*r/ 41.1%

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

    13. mul-1-neg 41.1%

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

    14. sqr-neg 41.1%

        \[\leadsto \left|ew \cdot \frac{1}{\sqrt{1 + \color{blue}{\frac{\tan t \cdot eh}{ew} \cdot \frac{\tan t \cdot eh}{ew}}}} - 0\right| \]

14. Simplified 41.1%

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

15. Final simplification 41.1%

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

Alternative 13: 41.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Cos[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $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. sin-mult 62.6%

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

   2. associate-*r/ 62.6%

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

5. Applied egg-rr 60.4%

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

   1. +-inverses 60.4%

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

   2. *-commutative 60.4%

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

   3. associate-/l* 60.4%

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

   4. div0 60.4%

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

7. Simplified 60.4%

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

8. Taylor expanded in t around 0 41.4%

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

   1. *-commutative 41.4%

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

   2. mul-1-neg 41.4%

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

   3. associate-*r/ 41.4%

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

   4. distribute-rgt-neg-in 41.4%

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

   5. distribute-neg-frac 41.4%

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

10. Simplified 41.4%

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

11. Taylor expanded in t around 0 39.8%

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

    1. associate-*r/ 39.8%

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

    2. *-commutative 39.8%

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

    3. associate-*r* 39.8%

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

    4. neg-mul-1 39.8%

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

13. Simplified 39.8%

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

14. Final simplification 39.8%

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

Alternative 14: 40.3% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(t * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $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. sin-mult 62.6%

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

   2. associate-*r/ 62.6%

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

5. Applied egg-rr 60.4%

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

   1. +-inverses 60.4%

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

   2. *-commutative 60.4%

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

   3. associate-/l* 60.4%

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

   4. div0 60.4%

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

7. Simplified 60.4%

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

8. Taylor expanded in t around 0 41.4%

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

   1. *-commutative 41.4%

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

   2. mul-1-neg 41.4%

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

   3. associate-*r/ 41.4%

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

   4. distribute-rgt-neg-in 41.4%

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

   5. distribute-neg-frac 41.4%

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

10. Simplified 41.4%

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

11. Taylor expanded in t around 0 39.8%

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

    1. associate-*r/ 39.8%

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

    2. *-commutative 39.8%

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

    3. associate-*r* 39.8%

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

    4. neg-mul-1 39.8%

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

13. Simplified 39.8%

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

    1. cos-atan 38.9%

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

    2. associate-/l* 38.8%

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

    3. associate-/l* 38.9%

       \[\leadsto \left|ew \cdot \frac{1}{\sqrt{1 + \frac{-eh}{\frac{ew}{t}} \cdot \color{blue}{\frac{-eh}{\frac{ew}{t}}}}} - 0\right| \]

15. Applied egg-rr 38.9%

    \[\leadsto \left|ew \cdot \color{blue}{\frac{1}{\sqrt{1 + \frac{-eh}{\frac{ew}{t}} \cdot \frac{-eh}{\frac{ew}{t}}}}} - 0\right| \]
16. Step-by-step derivation

    1. distribute-frac-neg 38.9%

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

    2. distribute-frac-neg 38.9%

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

    3. sqr-neg 38.9%

       \[\leadsto \left|ew \cdot \frac{1}{\sqrt{1 + \color{blue}{\frac{eh}{\frac{ew}{t}} \cdot \frac{eh}{\frac{ew}{t}}}}} - 0\right| \]

    4. associate-/l* 38.8%

       \[\leadsto \left|ew \cdot \frac{1}{\sqrt{1 + \color{blue}{\frac{eh \cdot t}{ew}} \cdot \frac{eh}{\frac{ew}{t}}}} - 0\right| \]

    5. *-commutative 38.8%

       \[\leadsto \left|ew \cdot \frac{1}{\sqrt{1 + \frac{\color{blue}{t \cdot eh}}{ew} \cdot \frac{eh}{\frac{ew}{t}}}} - 0\right| \]

    6. associate-/l* 38.9%

       \[\leadsto \left|ew \cdot \frac{1}{\sqrt{1 + \color{blue}{\frac{t}{\frac{ew}{eh}}} \cdot \frac{eh}{\frac{ew}{t}}}} - 0\right| \]

    7. associate-/l* 38.8%

       \[\leadsto \left|ew \cdot \frac{1}{\sqrt{1 + \frac{t}{\frac{ew}{eh}} \cdot \color{blue}{\frac{eh \cdot t}{ew}}}} - 0\right| \]

    8. *-commutative 38.8%

       \[\leadsto \left|ew \cdot \frac{1}{\sqrt{1 + \frac{t}{\frac{ew}{eh}} \cdot \frac{\color{blue}{t \cdot eh}}{ew}}} - 0\right| \]

    9. associate-/l* 38.9%

       \[\leadsto \left|ew \cdot \frac{1}{\sqrt{1 + \frac{t}{\frac{ew}{eh}} \cdot \color{blue}{\frac{t}{\frac{ew}{eh}}}}} - 0\right| \]

    10. hypot-1-def 39.0%

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

    11. associate-/l* 38.9%

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

    12. *-rgt-identity 38.9%

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

    13. associate-*r/ 38.9%

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

    14. associate-*l* 39.0%

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

    15. associate-*r/ 39.0%

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

    16. *-rgt-identity 39.0%

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

17. Simplified 39.0%

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

18. Final simplification 39.0%

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

Reproduce

herbie shell --seed 2023178 
(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)))))))