Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.4s

Alternatives: 11

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

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-eh\right) \cdot \frac{\tan t}{ew}\\ \left|\mathsf{fma}\left(\frac{\cos t}{\sqrt{1 + {t\_1}^{2}}}, ew, \sin \tan^{-1} t\_1 \cdot \left(-eh \cdot \sin t\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (- eh) (/ (tan t) ew))))
   (fabs
    (fma
     (/ (cos t) (sqrt (+ 1.0 (pow t_1 2.0))))
     ew
     (* (sin (atan t_1)) (- (* eh (sin t))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh * (tan(t) / ew);
	return fabs(fma((cos(t) / sqrt((1.0 + pow(t_1, 2.0)))), ew, (sin(atan(t_1)) * -(eh * sin(t)))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) * Float64(tan(t) / ew))
	return abs(fma(Float64(cos(t) / sqrt(Float64(1.0 + (t_1 ^ 2.0)))), ew, Float64(sin(atan(t_1)) * Float64(-Float64(eh * sin(t))))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[Cos[t], $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision] * (-N[(eh * N[Sin[t], $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. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \left|\mathsf{fma}\left(\frac{\cos t}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}, ew, \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-eh \cdot \sin t\right)\right)\right| \]
6. Add Preprocessing

Alternative 2: 51.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot ew\\ t_2 := \tan^{-1} \left(-\frac{eh \cdot \tan t}{ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) ew)) (t_2 (atan (- (/ (* eh (tan t)) ew)))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) -2e-259)
     (/ 1.0 (fabs (/ 1.0 ew)))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * ew;
	double t_2 = atan(-((eh * tan(t)) / ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -2e-259) {
		tmp = 1.0 / fabs((1.0 / 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) :: t_2
    real(8) :: tmp
    t_1 = cos(t) * ew
    t_2 = atan(-((eh * tan(t)) / ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= (-2d-259)) then
        tmp = 1.0d0 / abs((1.0d0 / 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.cos(t) * ew;
	double t_2 = Math.atan(-((eh * Math.tan(t)) / ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= -2e-259) {
		tmp = 1.0 / Math.abs((1.0 / ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.cos(t) * ew
	t_2 = math.atan(-((eh * math.tan(t)) / ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= -2e-259:
		tmp = 1.0 / math.fabs((1.0 / ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * ew)
	t_2 = atan(Float64(-Float64(Float64(eh * tan(t)) / ew)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -2e-259)
		tmp = Float64(1.0 / abs(Float64(1.0 / ew)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = cos(t) * ew;
	t_2 = atan(-((eh * tan(t)) / ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -2e-259)
		tmp = 1.0 / abs((1.0 / ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[(-N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-259], N[(1.0 / N[Abs[N[(1.0 / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -2.0000000000000001e-259

   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 76.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 42.4

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   7. Applied rewrites 42.4%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   if -2.0000000000000001e-259 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) 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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 75.6%

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

   7. Taylor expanded in ew around inf

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos t \cdot ew} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos t \cdot ew} \]

      3. lower-cos.f64 61.2

         \[\leadsto \color{blue}{\cos t} \cdot ew \]

   9. Applied rewrites 61.2%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(-\frac{eh \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(-\frac{eh \cdot \tan t}{ew}\right) \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(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)))))))