Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 15.8s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ \left|\mathsf{fma}\left(\left(-eh\right) \cdot \sin t, -\sin t\_1, \left(\cos t \cdot ew\right) \cdot \cos t\_1\right)\right| \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $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. Add Preprocessing

3. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 99.2

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

5. Applied rewrites 99.2%

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

6. Final simplification 99.2%

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

Alternative 3: 85.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan t \cdot eh\\ t_2 := eh \cdot \sin t\\ t_3 := \left|t\_2\right|\\ \mathbf{if}\;eh \leq -1.15 \cdot 10^{+75}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq 4.1 \cdot 10^{+177}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(\frac{t\_2}{ew}, t\_1, \cos t \cdot ew\right)\right|}{\frac{--1}{{\left({\left(\frac{ew}{t\_1}\right)}^{-2} + 1\right)}^{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (tan t) eh)) (t_2 (* eh (sin t))) (t_3 (fabs t_2)))
   (if (<= eh -1.15e+75)
     t_3
     (if (<= eh 4.1e+177)
       (/
        (fabs (fma (/ t_2 ew) t_1 (* (cos t) ew)))
        (/ (- -1.0) (pow (+ (pow (/ ew t_1) -2.0) 1.0) -0.5)))
       t_3))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) * eh;
	double t_2 = eh * sin(t);
	double t_3 = fabs(t_2);
	double tmp;
	if (eh <= -1.15e+75) {
		tmp = t_3;
	} else if (eh <= 4.1e+177) {
		tmp = fabs(fma((t_2 / ew), t_1, (cos(t) * ew))) / (-(-1.0) / pow((pow((ew / t_1), -2.0) + 1.0), -0.5));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) * eh)
	t_2 = Float64(eh * sin(t))
	t_3 = abs(t_2)
	tmp = 0.0
	if (eh <= -1.15e+75)
		tmp = t_3;
	elseif (eh <= 4.1e+177)
		tmp = Float64(abs(fma(Float64(t_2 / ew), t_1, Float64(cos(t) * ew))) / Float64(Float64(-(-1.0)) / (Float64((Float64(ew / t_1) ^ -2.0) + 1.0) ^ -0.5)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[t$95$2], $MachinePrecision]}, If[LessEqual[eh, -1.15e+75], t$95$3, If[LessEqual[eh, 4.1e+177], N[(N[Abs[N[(N[(t$95$2 / ew), $MachinePrecision] * t$95$1 + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((--1.0) / N[Power[N[(N[Power[N[(ew / t$95$1), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.1499999999999999e75 or 4.10000000000000014e177 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 28.9%

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

   6. Applied rewrites 38.6%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 78.6

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

   9. Applied rewrites 78.6%

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

   if -1.1499999999999999e75 < eh < 4.10000000000000014e177

   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. Add Preprocessing

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 76.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. pow1/2 N/A

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

   8. Applied rewrites 91.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.15 \cdot 10^{+75}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{elif}\;eh \leq 4.1 \cdot 10^{+177}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(\frac{eh \cdot \sin t}{ew}, \tan t \cdot eh, \cos t \cdot ew\right)\right|}{\frac{--1}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := \left|\mathsf{fma}\left(\frac{t\_1}{ew}, \tan t \cdot eh, \cos t \cdot ew\right)\right| \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ \mathbf{if}\;ew \leq -5.8 \cdot 10^{-15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 8 \cdot 10^{-130}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t)))
        (t_2
         (*
          (fabs (fma (/ t_1 ew) (* (tan t) eh) (* (cos t) ew)))
          (cos (atan (* (/ (tan t) ew) eh))))))
   (if (<= ew -5.8e-15) t_2 (if (<= ew 8e-130) (fabs t_1) t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = fabs(fma((t_1 / ew), (tan(t) * eh), (cos(t) * ew))) * cos(atan(((tan(t) / ew) * eh)));
	double tmp;
	if (ew <= -5.8e-15) {
		tmp = t_2;
	} else if (ew <= 8e-130) {
		tmp = fabs(t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = Float64(abs(fma(Float64(t_1 / ew), Float64(tan(t) * eh), Float64(cos(t) * ew))) * cos(atan(Float64(Float64(tan(t) / ew) * eh))))
	tmp = 0.0
	if (ew <= -5.8e-15)
		tmp = t_2;
	elseif (ew <= 8e-130)
		tmp = abs(t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[N[(N[(t$95$1 / ew), $MachinePrecision] * N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -5.8e-15], t$95$2, If[LessEqual[ew, 8e-130], N[Abs[t$95$1], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.80000000000000037e-15 or 8.0000000000000007e-130 < 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. Add Preprocessing

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 83.8%

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

   if -5.80000000000000037e-15 < ew < 8.0000000000000007e-130

   1. Initial program 99.9%

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

   4. Applied rewrites 31.4%

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

   6. Applied rewrites 52.1%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 75.4

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

   9. Applied rewrites 75.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.8 \cdot 10^{-15}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh \cdot \sin t}{ew}, \tan t \cdot eh, \cos t \cdot ew\right)\right| \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ \mathbf{elif}\;ew \leq 8 \cdot 10^{-130}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh \cdot \sin t}{ew}, \tan t \cdot eh, \cos t \cdot ew\right)\right| \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(-ew\right) \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -5.8 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 8.5 \cdot 10^{-130}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (- ew) (cos t)))))
   (if (<= ew -5.8e-15) t_1 (if (<= ew 8.5e-130) (fabs (* eh (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((-ew * cos(t)));
	double tmp;
	if (ew <= -5.8e-15) {
		tmp = t_1;
	} else if (ew <= 8.5e-130) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = 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 = abs((-ew * cos(t)))
    if (ew <= (-5.8d-15)) then
        tmp = t_1
    else if (ew <= 8.5d-130) then
        tmp = abs((eh * sin(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((-ew * Math.cos(t)));
	double tmp;
	if (ew <= -5.8e-15) {
		tmp = t_1;
	} else if (ew <= 8.5e-130) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((-ew * math.cos(t)))
	tmp = 0
	if ew <= -5.8e-15:
		tmp = t_1
	elif ew <= 8.5e-130:
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(-ew) * cos(t)))
	tmp = 0.0
	if (ew <= -5.8e-15)
		tmp = t_1;
	elseif (ew <= 8.5e-130)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((-ew * cos(t)));
	tmp = 0.0;
	if (ew <= -5.8e-15)
		tmp = t_1;
	elseif (ew <= 8.5e-130)
		tmp = abs((eh * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -5.8e-15], t$95$1, If[LessEqual[ew, 8.5e-130], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.80000000000000037e-15 or 8.50000000000000033e-130 < 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. Add Preprocessing

   4. Applied rewrites 83.8%

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

   6. Applied rewrites 91.2%

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

   7. Taylor expanded in ew around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(ew\right)\right)} \cdot \cos t\right| \]

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 82.2

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

   9. Applied rewrites 82.2%

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

   if -5.80000000000000037e-15 < ew < 8.50000000000000033e-130

   1. Initial program 99.9%

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

   4. Applied rewrites 31.4%

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

   6. Applied rewrites 52.1%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 75.4

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

   9. Applied rewrites 75.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.8 \cdot 10^{-15}:\\ \;\;\;\;\left|\left(-ew\right) \cdot \cos t\right|\\ \mathbf{elif}\;ew \leq 8.5 \cdot 10^{-130}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-ew\right) \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-94}:\\ \;\;\;\;\left|-ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= t -1.4e-24) t_1 (if (<= t 6.5e-94) (fabs (- ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (t <= -1.4e-24) {
		tmp = t_1;
	} else if (t <= 6.5e-94) {
		tmp = fabs(-ew);
	} else {
		tmp = 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 = abs((eh * sin(t)))
    if (t <= (-1.4d-24)) then
        tmp = t_1
    else if (t <= 6.5d-94) then
        tmp = abs(-ew)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (t <= -1.4e-24) {
		tmp = t_1;
	} else if (t <= 6.5e-94) {
		tmp = Math.abs(-ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if t <= -1.4e-24:
		tmp = t_1
	elif t <= 6.5e-94:
		tmp = math.fabs(-ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (t <= -1.4e-24)
		tmp = t_1;
	elseif (t <= 6.5e-94)
		tmp = abs(Float64(-ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (t <= -1.4e-24)
		tmp = t_1;
	elseif (t <= 6.5e-94)
		tmp = abs(-ew);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.4e-24], t$95$1, If[LessEqual[t, 6.5e-94], N[Abs[(-ew)], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.4000000000000001e-24 or 6.4999999999999996e-94 < t

   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. Add Preprocessing

   4. Applied rewrites 51.2%

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

   6. Applied rewrites 67.0%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 55.4

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

   9. Applied rewrites 55.4%

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

   if -1.4000000000000001e-24 < t < 6.4999999999999996e-94

   1. Initial program 100.0%

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

   4. Applied rewrites 78.9%

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

   6. Applied rewrites 87.4%

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

   7. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left|\color{blue}{\mathsf{neg}\left(ew\right)}\right| \]

      2. lower-neg.f64 78.6

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

   9. Applied rewrites 78.6%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-24}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-94}:\\ \;\;\;\;\left|-ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.3% accurate, 172.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 61.3%

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

6. Applied rewrites 74.4%

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

7. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

      \[\leadsto \left|\color{blue}{\mathsf{neg}\left(ew\right)}\right| \]

   2. lower-neg.f64 40.2

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

9. Applied rewrites 40.2%

   \[\leadsto \left|\color{blue}{-ew}\right| \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024267 
(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)))))))