Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.3s

Alternatives: 9

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot eh + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{fma}\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), eh, \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

   5. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), eh, \left(\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), \left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

Alternative 3: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\color{blue}{\left(\frac{eh}{ew \cdot t}\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(ew \cdot t\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(t \cdot ew\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f64 99.2%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(t, ew\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot eh + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{fma}\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), eh, \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

   5. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), eh, \left(\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eh around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right) + ew \cdot \sin t\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right), \left(ew \cdot \sin t\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right), \left(ew \cdot \sin t\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(eh \cdot \cos t\right), \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right), \left(ew \cdot \sin t\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \cos t\right), \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right), \left(ew \cdot \sin t\right)\right)\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right), \left(ew \cdot \sin t\right)\right)\right) \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right), \left(ew \cdot \sin t\right)\right)\right) \]

   7. atan-lowering-atan.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right), \left(ew \cdot \sin t\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(ew \cdot \tan t\right)\right)\right)\right)\right), \left(ew \cdot \sin t\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \tan t\right)\right)\right)\right)\right), \left(ew \cdot \sin t\right)\right)\right) \]

   10. tan-lowering-tan.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right), \left(ew \cdot \sin t\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(ew, \sin t\right)\right)\right) \]

   12. sin-lowering-sin.f64 98.4%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right)\right)\right) \]

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Alternative 5: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\mathsf{fma}\left(1, eh, \sin t \cdot \frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\right)\right|\\ \mathbf{if}\;ew \leq -6.8 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\left|\cos t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (fma 1.0 eh (* (sin t) (/ ew (hypot 1.0 (/ eh (* ew (tan t))))))))))
   (if (<= ew -6.8e-38) t_1 (if (<= ew 5e-34) (fabs (* (cos t) eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(fma(1.0, eh, (sin(t) * (ew / hypot(1.0, (eh / (ew * tan(t))))))));
	double tmp;
	if (ew <= -6.8e-38) {
		tmp = t_1;
	} else if (ew <= 5e-34) {
		tmp = fabs((cos(t) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(fma(1.0, eh, Float64(sin(t) * Float64(ew / hypot(1.0, Float64(eh / Float64(ew * tan(t))))))))
	tmp = 0.0
	if (ew <= -6.8e-38)
		tmp = t_1;
	elseif (ew <= 5e-34)
		tmp = abs(Float64(cos(t) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(1.0 * eh + N[(N[Sin[t], $MachinePrecision] * N[(ew / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -6.8e-38], t$95$1, If[LessEqual[ew, 5e-34], N[Abs[N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -6.8000000000000004e-38 or 5.0000000000000003e-34 < ew

   1. Initial program 99.8%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot eh + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      4. fma-define N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{fma}\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), eh, \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), eh, \left(\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

      1. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{eh \cdot \frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(eh \cdot \frac{\frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \left(\frac{\frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\left(\frac{1}{ew \cdot \tan t}\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\left(\frac{\frac{1}{ew}}{\tan t}\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{ew}\right), \tan t\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \tan t\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. hypot-undefine N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. associate-/r* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 89.3%

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

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\color{blue}{1}, eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 90.5%

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

   if -6.8000000000000004e-38 < ew < 5.0000000000000003e-34

   1. Initial program 99.8%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot eh + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      4. fma-define N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{fma}\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), eh, \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), eh, \left(\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

      1. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{eh \cdot \frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(eh \cdot \frac{\frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \left(\frac{\frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\left(\frac{1}{ew \cdot \tan t}\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\left(\frac{\frac{1}{ew}}{\tan t}\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{ew}\right), \tan t\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \tan t\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. hypot-undefine N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. associate-/r* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 50.7%

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

   7. Taylor expanded in eh around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \cos t\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \cos t\right)\right) \]

      2. cos-lowering-cos.f64 89.8%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right)\right) \]

   9. Simplified 89.8%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6.8 \cdot 10^{-38}:\\ \;\;\;\;\left|\mathsf{fma}\left(1, eh, \sin t \cdot \frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\right)\right|\\ \mathbf{elif}\;ew \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\left|\cos t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(1, eh, \sin t \cdot \frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\cos t \cdot eh\right|\\ \mathbf{if}\;eh \leq -1.92 \cdot 10^{-134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.4 \cdot 10^{-90}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (cos t) eh))))
   (if (<= eh -1.92e-134) t_1 (if (<= eh 2.4e-90) (fabs (* ew (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((cos(t) * eh));
	double tmp;
	if (eh <= -1.92e-134) {
		tmp = t_1;
	} else if (eh <= 2.4e-90) {
		tmp = fabs((ew * 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((cos(t) * eh))
    if (eh <= (-1.92d-134)) then
        tmp = t_1
    else if (eh <= 2.4d-90) then
        tmp = abs((ew * 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((Math.cos(t) * eh));
	double tmp;
	if (eh <= -1.92e-134) {
		tmp = t_1;
	} else if (eh <= 2.4e-90) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.cos(t) * eh))
	tmp = 0
	if eh <= -1.92e-134:
		tmp = t_1
	elif eh <= 2.4e-90:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(cos(t) * eh))
	tmp = 0.0
	if (eh <= -1.92e-134)
		tmp = t_1;
	elseif (eh <= 2.4e-90)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((cos(t) * eh));
	tmp = 0.0;
	if (eh <= -1.92e-134)
		tmp = t_1;
	elseif (eh <= 2.4e-90)
		tmp = abs((ew * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.92e-134], t$95$1, If[LessEqual[eh, 2.4e-90], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.9199999999999999e-134 or 2.4000000000000002e-90 < eh

   1. Initial program 99.8%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot eh + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      4. fma-define N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{fma}\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), eh, \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), eh, \left(\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

      1. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{eh \cdot \frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(eh \cdot \frac{\frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \left(\frac{\frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\left(\frac{1}{ew \cdot \tan t}\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\left(\frac{\frac{1}{ew}}{\tan t}\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{ew}\right), \tan t\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \tan t\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. hypot-undefine N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. associate-/r* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 64.2%

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

   7. Taylor expanded in eh around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \cos t\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \cos t\right)\right) \]

      2. cos-lowering-cos.f64 80.1%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right)\right) \]

   9. Simplified 80.1%

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

   if -1.9199999999999999e-134 < eh < 2.4000000000000002e-90

   1. Initial program 99.8%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot eh + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      4. fma-define N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{fma}\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), eh, \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), eh, \left(\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \sin t\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \sin t\right)\right) \]

      2. sin-lowering-sin.f64 74.3%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right)\right) \]

   7. Simplified 74.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.92 \cdot 10^{-134}:\\ \;\;\;\;\left|\cos t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 2.4 \cdot 10^{-90}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\cos t \cdot eh\right|\\ \mathbf{if}\;eh \leq -1.62 \cdot 10^{-211}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.15 \cdot 10^{-266}:\\ \;\;\;\;\left|t \cdot \left(ew + ew \cdot \left(\left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(0.008333333333333333 + \left(t \cdot t\right) \cdot -0.0001984126984126984\right) + -0.16666666666666666\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (cos t) eh))))
   (if (<= eh -1.62e-211)
     t_1
     (if (<= eh 2.15e-266)
       (fabs
        (*
         t
         (+
          ew
          (*
           ew
           (*
            (* t t)
            (+
             (*
              (* t t)
              (+ 0.008333333333333333 (* (* t t) -0.0001984126984126984)))
             -0.16666666666666666))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((cos(t) * eh));
	double tmp;
	if (eh <= -1.62e-211) {
		tmp = t_1;
	} else if (eh <= 2.15e-266) {
		tmp = fabs((t * (ew + (ew * ((t * t) * (((t * t) * (0.008333333333333333 + ((t * t) * -0.0001984126984126984))) + -0.16666666666666666))))));
	} 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((cos(t) * eh))
    if (eh <= (-1.62d-211)) then
        tmp = t_1
    else if (eh <= 2.15d-266) then
        tmp = abs((t * (ew + (ew * ((t * t) * (((t * t) * (0.008333333333333333d0 + ((t * t) * (-0.0001984126984126984d0)))) + (-0.16666666666666666d0)))))))
    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((Math.cos(t) * eh));
	double tmp;
	if (eh <= -1.62e-211) {
		tmp = t_1;
	} else if (eh <= 2.15e-266) {
		tmp = Math.abs((t * (ew + (ew * ((t * t) * (((t * t) * (0.008333333333333333 + ((t * t) * -0.0001984126984126984))) + -0.16666666666666666))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.cos(t) * eh))
	tmp = 0
	if eh <= -1.62e-211:
		tmp = t_1
	elif eh <= 2.15e-266:
		tmp = math.fabs((t * (ew + (ew * ((t * t) * (((t * t) * (0.008333333333333333 + ((t * t) * -0.0001984126984126984))) + -0.16666666666666666))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(cos(t) * eh))
	tmp = 0.0
	if (eh <= -1.62e-211)
		tmp = t_1;
	elseif (eh <= 2.15e-266)
		tmp = abs(Float64(t * Float64(ew + Float64(ew * Float64(Float64(t * t) * Float64(Float64(Float64(t * t) * Float64(0.008333333333333333 + Float64(Float64(t * t) * -0.0001984126984126984))) + -0.16666666666666666))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((cos(t) * eh));
	tmp = 0.0;
	if (eh <= -1.62e-211)
		tmp = t_1;
	elseif (eh <= 2.15e-266)
		tmp = abs((t * (ew + (ew * ((t * t) * (((t * t) * (0.008333333333333333 + ((t * t) * -0.0001984126984126984))) + -0.16666666666666666))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.62e-211], t$95$1, If[LessEqual[eh, 2.15e-266], N[Abs[N[(t * N[(ew + N[(ew * N[(N[(t * t), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(t * t), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.61999999999999999e-211 or 2.15000000000000014e-266 < eh

   1. Initial program 99.8%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot eh + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      4. fma-define N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{fma}\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), eh, \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), eh, \left(\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

      1. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(\frac{eh \cdot \frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \left(eh \cdot \frac{\frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \left(\frac{\frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\left(\frac{1}{ew \cdot \tan t}\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\left(\frac{\frac{1}{ew}}{\tan t}\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{ew}\right), \tan t\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \tan t\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. hypot-undefine N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. associate-/r* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right)\right)\right), eh, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 71.1%

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

   7. Taylor expanded in eh around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \cos t\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \cos t\right)\right) \]

      2. cos-lowering-cos.f64 69.6%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right)\right) \]

   9. Simplified 69.6%

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

   if -1.61999999999999999e-211 < eh < 2.15000000000000014e-266

   1. Initial program 99.9%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot eh + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      4. fma-define N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{fma}\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), eh, \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), eh, \left(\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \sin t\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \sin t\right)\right) \]

      2. sin-lowering-sin.f64 89.2%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right)\right) \]

   7. Simplified 89.2%

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

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(t \cdot \left(ew + {t}^{2} \cdot \left(\frac{-1}{6} \cdot ew + {t}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(ew \cdot {t}^{2}\right) + \frac{1}{120} \cdot ew\right)\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \left(ew + {t}^{2} \cdot \left(\frac{-1}{6} \cdot ew + {t}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(ew \cdot {t}^{2}\right) + \frac{1}{120} \cdot ew\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \left({t}^{2} \cdot \left(\frac{-1}{6} \cdot ew + {t}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(ew \cdot {t}^{2}\right) + \frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\left({t}^{2}\right), \left(\frac{-1}{6} \cdot ew + {t}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(ew \cdot {t}^{2}\right) + \frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\frac{-1}{6} \cdot ew + {t}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(ew \cdot {t}^{2}\right) + \frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{-1}{6} \cdot ew + {t}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(ew \cdot {t}^{2}\right) + \frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\left(\frac{-1}{6} \cdot ew\right), \left({t}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(ew \cdot {t}^{2}\right) + \frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\left(ew \cdot \frac{-1}{6}\right), \left({t}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(ew \cdot {t}^{2}\right) + \frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(ew, \frac{-1}{6}\right), \left({t}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(ew \cdot {t}^{2}\right) + \frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(ew, \frac{-1}{6}\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \left(\frac{-1}{5040} \cdot \left(ew \cdot {t}^{2}\right) + \frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(ew, \frac{-1}{6}\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\frac{-1}{5040} \cdot \left(ew \cdot {t}^{2}\right) + \frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(ew, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{-1}{5040} \cdot \left(ew \cdot {t}^{2}\right) + \frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(ew, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\left(\frac{-1}{5040} \cdot \left(ew \cdot {t}^{2}\right)\right), \left(\frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(ew, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{5040}, \left(ew \cdot {t}^{2}\right)\right), \left(\frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(ew, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(ew, \left({t}^{2}\right)\right)\right), \left(\frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(ew, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(ew, \left(t \cdot t\right)\right)\right), \left(\frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(ew, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(\frac{1}{120} \cdot ew\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(ew, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(ew \cdot \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 49.0%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(ew, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(ew, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 49.0%

       \[\leadsto \left|\color{blue}{t \cdot \left(ew + \left(t \cdot t\right) \cdot \left(ew \cdot -0.16666666666666666 + \left(t \cdot t\right) \cdot \left(-0.0001984126984126984 \cdot \left(ew \cdot \left(t \cdot t\right)\right) + ew \cdot 0.008333333333333333\right)\right)\right)}\right| \]

   11. Taylor expanded in ew around 0

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \color{blue}{\left(ew \cdot \left({t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \left({t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\left({t}^{2}\right), \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\left(t \cdot t\right), \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({t}^{2}\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(t \cdot t\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{-1}{5040} \cdot {t}^{2}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({t}^{2} \cdot \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({t}^{2}\right), \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(t \cdot t\right), \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 49.0%

           \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(ew, \mathsf{*.f64}\left(ew, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 49.0%

       \[\leadsto \left|t \cdot \left(ew + \color{blue}{ew \cdot \left(\left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(0.008333333333333333 + \left(t \cdot t\right) \cdot -0.0001984126984126984\right) + -0.16666666666666666\right)\right)}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.62 \cdot 10^{-211}:\\ \;\;\;\;\left|\cos t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 2.15 \cdot 10^{-266}:\\ \;\;\;\;\left|t \cdot \left(ew + ew \cdot \left(\left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(0.008333333333333333 + \left(t \cdot t\right) \cdot -0.0001984126984126984\right) + -0.16666666666666666\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.16 \cdot 10^{-134}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{elif}\;eh \leq 3.6 \cdot 10^{-186}:\\ \;\;\;\;\left|t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -1.16e-134)
   (fabs eh)
   (if (<= eh 3.6e-186) (fabs (* t ew)) (fabs eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.16e-134) {
		tmp = fabs(eh);
	} else if (eh <= 3.6e-186) {
		tmp = fabs((t * ew));
	} else {
		tmp = fabs(eh);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-1.16d-134)) then
        tmp = abs(eh)
    else if (eh <= 3.6d-186) then
        tmp = abs((t * ew))
    else
        tmp = abs(eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.16e-134) {
		tmp = Math.abs(eh);
	} else if (eh <= 3.6e-186) {
		tmp = Math.abs((t * ew));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -1.16e-134:
		tmp = math.fabs(eh)
	elif eh <= 3.6e-186:
		tmp = math.fabs((t * ew))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -1.16e-134)
		tmp = abs(eh);
	elseif (eh <= 3.6e-186)
		tmp = abs(Float64(t * ew));
	else
		tmp = abs(eh);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -1.16e-134)
		tmp = abs(eh);
	elseif (eh <= 3.6e-186)
		tmp = abs((t * ew));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -1.16e-134], N[Abs[eh], $MachinePrecision], If[LessEqual[eh, 3.6e-186], N[Abs[N[(t * ew), $MachinePrecision]], $MachinePrecision], N[Abs[eh], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.1600000000000001e-134 or 3.5999999999999998e-186 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right) \]

      3. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(ew \cdot \tan t\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \tan t\right)\right)\right)\right)\right)\right) \]

      6. tan-lowering-tan.f64 51.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 51.5%

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

      1. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{eh \cdot \frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(eh \cdot \frac{\frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \left(\frac{\frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\left(\frac{1}{ew \cdot \tan t}\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\left(\frac{\frac{1}{ew}}{\tan t}\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{ew}\right), \tan t\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \tan t\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right)\right) \]

      11. hypot-undefine N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right)\right) \]

      13. associate-/r* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 25.1%

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

   8. Taylor expanded in eh around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{eh}\right) \]
   9. Step-by-step derivation

   10. Simplified 51.9%

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

   if -1.1600000000000001e-134 < eh < 3.5999999999999998e-186

   1. Initial program 99.8%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot eh + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      4. fma-define N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{fma}\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), eh, \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), eh, \left(\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \sin t\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \sin t\right)\right) \]

      2. sin-lowering-sin.f64 75.1%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right)\right) \]

   7. Simplified 75.1%

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

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot t\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(t \cdot ew\right)\right) \]

      2. *-lowering-*.f64 42.2%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, ew\right)\right) \]

   10. Simplified 42.2%

       \[\leadsto \left|\color{blue}{t \cdot ew}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 42.6% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right) \]

   2. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right) \]

   3. atan-lowering-atan.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(ew \cdot \tan t\right)\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \tan t\right)\right)\right)\right)\right)\right) \]

   6. tan-lowering-tan.f64 42.6%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 42.6%

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

   1. sin-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right) \]

   3. div-inv N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{eh \cdot \frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(eh \cdot \frac{\frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \left(\frac{\frac{1}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\left(\frac{1}{ew \cdot \tan t}\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right)\right) \]

   7. associate-/r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\left(\frac{\frac{1}{ew}}{\tan t}\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{ew}\right), \tan t\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \tan t\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right)\right) \]

   10. tan-lowering-tan.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \left(\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}\right)\right)\right)\right)\right) \]

   11. hypot-undefine N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \left(\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right)\right) \]

   12. hypot-lowering-hypot.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right)\right) \]

   13. associate-/r* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, ew\right), \mathsf{tan.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right)\right)\right)\right) \]

7. Applied egg-rr 21.3%

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

8. Taylor expanded in eh around inf

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{eh}\right) \]
9. Step-by-step derivation

10. Simplified 43.1%

    \[\leadsto \left|\color{blue}{eh}\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))