Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.1s

Alternatives: 13

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    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

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Tanh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision] * N[Cos[t], $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. fp-cancel-sub-sign-inv N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. *-commutative N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 88.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ t_2 := \sinh^{-1} t\_1\\ \mathbf{if}\;eh \leq -1.9 \cdot 10^{-13} \lor \neg \left(eh \leq 4.2 \cdot 10^{+23}\right):\\ \;\;\;\;\left|\left(\tanh t\_2 \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_1, eh, \sin t \cdot ew\right)}{\cosh t\_2}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)) (t_2 (asinh t_1)))
   (if (or (<= eh -1.9e-13) (not (<= eh 4.2e+23)))
     (fabs (* (* (tanh t_2) (cos t)) eh))
     (fabs (/ (fma (* (cos t) t_1) eh (* (sin t) ew)) (cosh t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double t_2 = asinh(t_1);
	double tmp;
	if ((eh <= -1.9e-13) || !(eh <= 4.2e+23)) {
		tmp = fabs(((tanh(t_2) * cos(t)) * eh));
	} else {
		tmp = fabs((fma((cos(t) * t_1), eh, (sin(t) * ew)) / cosh(t_2)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	t_2 = asinh(t_1)
	tmp = 0.0
	if ((eh <= -1.9e-13) || !(eh <= 4.2e+23))
		tmp = abs(Float64(Float64(tanh(t_2) * cos(t)) * eh));
	else
		tmp = abs(Float64(fma(Float64(cos(t) * t_1), eh, Float64(sin(t) * ew)) / cosh(t_2)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[ArcSinh[t$95$1], $MachinePrecision]}, If[Or[LessEqual[eh, -1.9e-13], N[Not[LessEqual[eh, 4.2e+23]], $MachinePrecision]], N[Abs[N[(N[(N[Tanh[t$95$2], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[t$95$2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.9e-13 or 4.2000000000000003e23 < 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 eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 89.5

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

   5. Applied rewrites 89.5%

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

   7. Applied rewrites 89.5%

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

   if -1.9e-13 < eh < 4.2000000000000003e23

   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

   4. Applied rewrites 92.4%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.9 \cdot 10^{-13} \lor \neg \left(eh \leq 4.2 \cdot 10^{+23}\right):\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.95 \cdot 10^{-73} \lor \neg \left(eh \leq 2.4 \cdot 10^{-157}\right):\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -1.95e-73) (not (<= eh 2.4e-157)))
   (fabs (* (* (tanh (asinh (/ (/ eh (tan t)) ew))) (cos t)) eh))
   (fabs (* (cos (atan (* (/ (cos t) ew) (/ eh (sin t))))) (* (sin t) ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.95e-73) || !(eh <= 2.4e-157)) {
		tmp = fabs(((tanh(asinh(((eh / tan(t)) / ew))) * cos(t)) * eh));
	} else {
		tmp = fabs((cos(atan(((cos(t) / ew) * (eh / sin(t))))) * (sin(t) * ew)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -1.95e-73) or not (eh <= 2.4e-157):
		tmp = math.fabs(((math.tanh(math.asinh(((eh / math.tan(t)) / ew))) * math.cos(t)) * eh))
	else:
		tmp = math.fabs((math.cos(math.atan(((math.cos(t) / ew) * (eh / math.sin(t))))) * (math.sin(t) * ew)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -1.95e-73) || !(eh <= 2.4e-157))
		tmp = abs(Float64(Float64(tanh(asinh(Float64(Float64(eh / tan(t)) / ew))) * cos(t)) * eh));
	else
		tmp = abs(Float64(cos(atan(Float64(Float64(cos(t) / ew) * Float64(eh / sin(t))))) * Float64(sin(t) * ew)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -1.95e-73) || ~((eh <= 2.4e-157)))
		tmp = abs(((tanh(asinh(((eh / tan(t)) / ew))) * cos(t)) * eh));
	else
		tmp = abs((cos(atan(((cos(t) / ew) * (eh / sin(t))))) * (sin(t) * ew)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -1.95e-73], N[Not[LessEqual[eh, 2.4e-157]], $MachinePrecision]], N[Abs[N[(N[(N[Tanh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.94999999999999991e-73 or 2.4e-157 < 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 eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 80.1

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

   5. Applied rewrites 80.1%

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

   7. Applied rewrites 80.1%

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

   if -1.94999999999999991e-73 < eh < 2.4e-157

   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 eh around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 82.9

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

   5. Applied rewrites 82.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.95 \cdot 10^{-73} \lor \neg \left(eh \leq 2.4 \cdot 10^{-157}\right):\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)\\ \mathbf{if}\;eh \leq -1.95 \cdot 10^{-73} \lor \neg \left(eh \leq 2.4 \cdot 10^{-157}\right):\\ \;\;\;\;\left|\left(\tanh t\_1 \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{ew \cdot \sin t}{\cosh t\_1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (asinh (/ (/ eh (tan t)) ew))))
   (if (or (<= eh -1.95e-73) (not (<= eh 2.4e-157)))
     (fabs (* (* (tanh t_1) (cos t)) eh))
     (fabs (/ (* ew (sin t)) (cosh t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = asinh(((eh / tan(t)) / ew));
	double tmp;
	if ((eh <= -1.95e-73) || !(eh <= 2.4e-157)) {
		tmp = fabs(((tanh(t_1) * cos(t)) * eh));
	} else {
		tmp = fabs(((ew * sin(t)) / cosh(t_1)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.asinh(((eh / math.tan(t)) / ew))
	tmp = 0
	if (eh <= -1.95e-73) or not (eh <= 2.4e-157):
		tmp = math.fabs(((math.tanh(t_1) * math.cos(t)) * eh))
	else:
		tmp = math.fabs(((ew * math.sin(t)) / math.cosh(t_1)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = asinh(Float64(Float64(eh / tan(t)) / ew))
	tmp = 0.0
	if ((eh <= -1.95e-73) || !(eh <= 2.4e-157))
		tmp = abs(Float64(Float64(tanh(t_1) * cos(t)) * eh));
	else
		tmp = abs(Float64(Float64(ew * sin(t)) / cosh(t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = asinh(((eh / tan(t)) / ew));
	tmp = 0.0;
	if ((eh <= -1.95e-73) || ~((eh <= 2.4e-157)))
		tmp = abs(((tanh(t_1) * cos(t)) * eh));
	else
		tmp = abs(((ew * sin(t)) / cosh(t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[eh, -1.95e-73], N[Not[LessEqual[eh, 2.4e-157]], $MachinePrecision]], N[Abs[N[(N[(N[Tanh[t$95$1], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] / N[Cosh[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.94999999999999991e-73 or 2.4e-157 < 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 eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 80.1

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

   5. Applied rewrites 80.1%

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

   7. Applied rewrites 80.1%

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

   if -1.94999999999999991e-73 < eh < 2.4e-157

   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

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 65.7

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

   5. Applied rewrites 65.7%

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

   7. Applied rewrites 63.5%

      \[\leadsto \left|\color{blue}{\frac{\frac{\frac{eh}{\tan t}}{ew} \cdot \mathsf{fma}\left(t \cdot t, -0.5 \cdot eh, eh\right) - \left(\left(-ew\right) \cdot \sin t\right) \cdot 1}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right| \]

   8. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 82.8

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

   10. Applied rewrites 82.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.95 \cdot 10^{-73} \lor \neg \left(eh \leq 2.4 \cdot 10^{-157}\right):\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{ew \cdot \sin t}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh \cdot eh}{ew}\\ t_2 := \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)\\ \mathbf{if}\;eh \leq 6.6 \cdot 10^{-308} \lor \neg \left(eh \leq 4.6 \cdot 10^{-161}\right):\\ \;\;\;\;\left|\left(\tanh t\_2 \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot t\_1 - \mathsf{fma}\left(0.3333333333333333, t\_1, -ew\right), t\_1\right)}{t}}{\cosh t\_2}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (* eh eh) ew)) (t_2 (asinh (/ (/ eh (tan t)) ew))))
   (if (or (<= eh 6.6e-308) (not (<= eh 4.6e-161)))
     (fabs (* (* (tanh t_2) (cos t)) eh))
     (fabs
      (/
       (/
        (fma (* t t) (- (* -0.5 t_1) (fma 0.3333333333333333 t_1 (- ew))) t_1)
        t)
       (cosh t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh * eh) / ew;
	double t_2 = asinh(((eh / tan(t)) / ew));
	double tmp;
	if ((eh <= 6.6e-308) || !(eh <= 4.6e-161)) {
		tmp = fabs(((tanh(t_2) * cos(t)) * eh));
	} else {
		tmp = fabs(((fma((t * t), ((-0.5 * t_1) - fma(0.3333333333333333, t_1, -ew)), t_1) / t) / cosh(t_2)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh * eh) / ew)
	t_2 = asinh(Float64(Float64(eh / tan(t)) / ew))
	tmp = 0.0
	if ((eh <= 6.6e-308) || !(eh <= 4.6e-161))
		tmp = abs(Float64(Float64(tanh(t_2) * cos(t)) * eh));
	else
		tmp = abs(Float64(Float64(fma(Float64(t * t), Float64(Float64(-0.5 * t_1) - fma(0.3333333333333333, t_1, Float64(-ew))), t_1) / t) / cosh(t_2)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[eh, 6.6e-308], N[Not[LessEqual[eh, 4.6e-161]], $MachinePrecision]], N[Abs[N[(N[(N[Tanh[t$95$2], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(t * t), $MachinePrecision] * N[(N[(-0.5 * t$95$1), $MachinePrecision] - N[(0.3333333333333333 * t$95$1 + (-ew)), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / t), $MachinePrecision] / N[Cosh[t$95$2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < 6.5999999999999996e-308 or 4.6e-161 < 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 eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 70.5%

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

   if 6.5999999999999996e-308 < eh < 4.6e-161

   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. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 78.6

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

   5. Applied rewrites 78.6%

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

   7. Applied rewrites 78.2%

      \[\leadsto \left|\color{blue}{\frac{\frac{\frac{eh}{\tan t}}{ew} \cdot \mathsf{fma}\left(t \cdot t, -0.5 \cdot eh, eh\right) - \left(\left(-ew\right) \cdot \sin t\right) \cdot 1}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\frac{\color{blue}{\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{eh}^{2}}{ew} - \left(-1 \cdot ew + \frac{1}{3} \cdot \frac{{eh}^{2}}{ew}\right)\right) + \frac{{eh}^{2}}{ew}}{t}}}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right| \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{\color{blue}{\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{eh}^{2}}{ew} - \left(-1 \cdot ew + \frac{1}{3} \cdot \frac{{eh}^{2}}{ew}\right)\right) + \frac{{eh}^{2}}{ew}}{t}}}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right| \]

   10. Applied rewrites 45.5%

       \[\leadsto \left|\frac{\color{blue}{\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh \cdot eh}{ew} - \mathsf{fma}\left(0.3333333333333333, \frac{eh \cdot eh}{ew}, -ew\right), \frac{eh \cdot eh}{ew}\right)}{t}}}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq 6.6 \cdot 10^{-308} \lor \neg \left(eh \leq 4.6 \cdot 10^{-161}\right):\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh \cdot eh}{ew} - \mathsf{fma}\left(0.3333333333333333, \frac{eh \cdot eh}{ew}, -ew\right), \frac{eh \cdot eh}{ew}\right)}{t}}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{-7} \lor \neg \left(t \leq 1.36 \cdot 10^{-8}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(-0.3333333333333333 \cdot \frac{eh \cdot t}{ew}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -2.05e-7) (not (<= t 1.36e-8)))
   (fabs
    (* (sin (atan (* -0.3333333333333333 (/ (* eh t) ew)))) (* (cos t) eh)))
   (fabs (* (tanh (asinh (/ (/ eh (tan t)) ew))) eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -2.05e-7) || !(t <= 1.36e-8)) {
		tmp = fabs((sin(atan((-0.3333333333333333 * ((eh * t) / ew)))) * (cos(t) * eh)));
	} else {
		tmp = fabs((tanh(asinh(((eh / tan(t)) / ew))) * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -2.05e-7) or not (t <= 1.36e-8):
		tmp = math.fabs((math.sin(math.atan((-0.3333333333333333 * ((eh * t) / ew)))) * (math.cos(t) * eh)))
	else:
		tmp = math.fabs((math.tanh(math.asinh(((eh / math.tan(t)) / ew))) * eh))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -2.05e-7) || !(t <= 1.36e-8))
		tmp = abs(Float64(sin(atan(Float64(-0.3333333333333333 * Float64(Float64(eh * t) / ew)))) * Float64(cos(t) * eh)));
	else
		tmp = abs(Float64(tanh(asinh(Float64(Float64(eh / tan(t)) / ew))) * eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -2.05e-7) || ~((t <= 1.36e-8)))
		tmp = abs((sin(atan((-0.3333333333333333 * ((eh * t) / ew)))) * (cos(t) * eh)));
	else
		tmp = abs((tanh(asinh(((eh / tan(t)) / ew))) * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -2.05e-7], N[Not[LessEqual[t, 1.36e-8]], $MachinePrecision]], N[Abs[N[(N[Sin[N[ArcTan[N[(-0.3333333333333333 * N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.05e-7 or 1.3599999999999999e-8 < t

   1. Initial program 99.6%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 39.3%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 55.6%

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

   if -2.05e-7 < t < 1.3599999999999999e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 72.6

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

   5. Applied rewrites 72.6%

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

   7. Applied rewrites 72.6%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{-7} \lor \neg \left(t \leq 1.36 \cdot 10^{-8}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(-0.3333333333333333 \cdot \frac{eh \cdot t}{ew}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.8% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin (atan (/ (* eh (/ (fma -0.3333333333333333 (* t t) 1.0) ew)) t)))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan(((eh * (fma(-0.3333333333333333, (t * t), 1.0) / ew)) / t))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(eh * Float64(fma(-0.3333333333333333, Float64(t * t), 1.0) / ew)) / t))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh * N[(N[(-0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-sin.f64 44.2

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

5. Applied rewrites 44.2%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 34.4%

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

9. Taylor expanded in eh around 0

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

11. Applied rewrites 44.4%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{eh \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, t \cdot t, 1\right)}{ew}}{t}\right) \cdot eh\right| \]
12. Add Preprocessing

Alternative 8: 42.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((fma((eh * t), -0.3333333333333333, (eh / t)) / ew))) * eh));
}

Julia

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

Wolfram

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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-sin.f64 44.2

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

5. Applied rewrites 44.2%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 34.4%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\frac{eh}{ew} \cdot -0.3333333333333333, t \cdot t, \frac{eh}{ew}\right)}{t}\right) \cdot eh\right| \]
9. Step-by-step derivation

10. Applied rewrites 22.8%

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

11. Taylor expanded in ew around 0

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

13. Applied rewrites 44.4%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(eh \cdot t, -0.3333333333333333, \frac{eh}{t}\right)}{ew}\right) \cdot eh\right| \]
14. Add Preprocessing

Alternative 9: 40.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-sin.f64 44.2

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

5. Applied rewrites 44.2%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 34.4%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 42.8%

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

Alternative 10: 22.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left|\tanh \sinh^{-1} \left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (* (tanh (asinh (* (* -0.3333333333333333 (/ eh ew)) t))) eh)))

C

double code(double eh, double ew, double t) {
	return fabs((tanh(asinh(((-0.3333333333333333 * (eh / ew)) * t))) * eh));
}

Python

def code(eh, ew, t):
	return math.fabs((math.tanh(math.asinh(((-0.3333333333333333 * (eh / ew)) * t))) * eh))

Julia

function code(eh, ew, t)
	return abs(Float64(tanh(asinh(Float64(Float64(-0.3333333333333333 * Float64(eh / ew)) * t))) * eh))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((tanh(asinh(((-0.3333333333333333 * (eh / ew)) * t))) * eh));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(-0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-sin.f64 44.2

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

5. Applied rewrites 44.2%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 34.4%

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

9. Taylor expanded in t around inf

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

11. Applied rewrites 26.8%

    \[\leadsto \left|\sin \tan^{-1} \left(-0.3333333333333333 \cdot \frac{eh \cdot t}{ew}\right) \cdot eh\right| \]
12. Step-by-step derivation

13. Applied rewrites 26.8%

    \[\leadsto \color{blue}{\left|\tanh \sinh^{-1} \left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t\right) \cdot eh\right|} \]

14. Final simplification 26.8%

    \[\leadsto \left|\tanh \sinh^{-1} \left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t\right) \cdot eh\right| \]
15. Add Preprocessing

Alternative 11: 22.5% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((-0.3333333333333333 * ((eh * t) / ew)))) * eh));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((sin(atan(((-0.3333333333333333d0) * ((eh * t) / ew)))) * eh))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(-0.3333333333333333 * N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-sin.f64 44.2

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

5. Applied rewrites 44.2%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 34.4%

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

9. Taylor expanded in t around inf

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

11. Applied rewrites 26.8%

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

12. Final simplification 26.8%

    \[\leadsto \left|\sin \tan^{-1} \left(-0.3333333333333333 \cdot \frac{eh \cdot t}{ew}\right) \cdot eh\right| \]
13. Add Preprocessing

Alternative 12: 22.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((-0.3333333333333333 * ((t / ew) * eh)))) * eh));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((sin(atan(((-0.3333333333333333d0) * ((t / ew) * eh)))) * eh))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(-0.3333333333333333 * N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-sin.f64 44.2

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

5. Applied rewrites 44.2%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 34.4%

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

9. Taylor expanded in t around inf

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

11. Applied rewrites 26.8%

    \[\leadsto \left|\sin \tan^{-1} \left(-0.3333333333333333 \cdot \frac{eh \cdot t}{ew}\right) \cdot eh\right| \]
12. Step-by-step derivation

13. Applied rewrites 26.7%

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

14. Final simplification 26.7%

    \[\leadsto \left|\sin \tan^{-1} \left(-0.3333333333333333 \cdot \left(\frac{t}{ew} \cdot eh\right)\right) \cdot eh\right| \]
15. Add Preprocessing

Alternative 13: 12.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \tanh \sinh^{-1} \left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t\right) \cdot eh \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (* (tanh (asinh (* (* -0.3333333333333333 (/ eh ew)) t))) eh))

C

double code(double eh, double ew, double t) {
	return tanh(asinh(((-0.3333333333333333 * (eh / ew)) * t))) * eh;
}

Python

def code(eh, ew, t):
	return math.tanh(math.asinh(((-0.3333333333333333 * (eh / ew)) * t))) * eh

Julia

function code(eh, ew, t)
	return Float64(tanh(asinh(Float64(Float64(-0.3333333333333333 * Float64(eh / ew)) * t))) * eh)
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = tanh(asinh(((-0.3333333333333333 * (eh / ew)) * t))) * eh;
end

Wolfram

code[eh_, ew_, t_] := N[(N[Tanh[N[ArcSinh[N[(N[(-0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-sin.f64 44.2

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

5. Applied rewrites 44.2%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 34.4%

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

9. Taylor expanded in t around inf

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

11. Applied rewrites 26.8%

    \[\leadsto \left|\sin \tan^{-1} \left(-0.3333333333333333 \cdot \frac{eh \cdot t}{ew}\right) \cdot eh\right| \]
12. Step-by-step derivation

    1. lift-fabs.f64 N/A

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

    2. rem-sqrt-square-rev N/A

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

13. Applied rewrites 14.0%

    \[\leadsto \color{blue}{\tanh \sinh^{-1} \left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t\right) \cdot eh} \]

14. Final simplification 14.0%

    \[\leadsto \tanh \sinh^{-1} \left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t\right) \cdot eh \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(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))))))))