Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.4s

Alternatives: 12

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

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

FPCore

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

C

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

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((((eh * cos(t)) * sin(atan((eh / (tan(t) * ew))))) + ((ew * sin(t)) * cos(atan(((eh / ew) / tan(t)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 29.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 + \left(eh \cdot \cos t\right) \cdot \sin t\_2 \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \left(t \cdot t\right) \cdot ew, -0.16666666666666666 \cdot ew\right), t \cdot t, ew\right) \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))) (t_2 (atan (/ (/ eh ew) (tan t)))))
   (if (<= (+ (* t_1 (cos t_2)) (* (* eh (cos t)) (sin t_2))) -2e-294)
     (fabs
      (*
       (fma
        (fma 0.008333333333333333 (* (* t t) ew) (* -0.16666666666666666 ew))
        (* t t)
        ew)
       t))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = atan(((eh / ew) / tan(t)));
	double tmp;
	if (((t_1 * cos(t_2)) + ((eh * cos(t)) * sin(t_2))) <= -2e-294) {
		tmp = fabs((fma(fma(0.008333333333333333, ((t * t) * ew), (-0.16666666666666666 * ew)), (t * t), ew) * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = atan(Float64(Float64(eh / ew) / tan(t)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) + Float64(Float64(eh * cos(t)) * sin(t_2))) <= -2e-294)
		tmp = abs(Float64(fma(fma(0.008333333333333333, Float64(Float64(t * t) * ew), Float64(-0.16666666666666666 * ew)), Float64(t * t), ew) * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-294], N[Abs[N[(N[(N[(0.008333333333333333 * N[(N[(t * t), $MachinePrecision] * ew), $MachinePrecision] + N[(-0.16666666666666666 * ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))) < -2.00000000000000003e-294

   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 58.5%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 37.9

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

   7. Applied rewrites 37.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 19.0%

       \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \left(t \cdot t\right) \cdot ew, -0.16666666666666666 \cdot ew\right), t \cdot t, ew\right) \cdot \color{blue}{t}\right| \]

   if -2.00000000000000003e-294 < (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))

   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 67.4%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 45.2

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

   7. Applied rewrites 45.2%

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

      1. lift-fabs.f64 N/A

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

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

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \sin t\right) \cdot \left(ew \cdot \sin t\right)}} \]

      3. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{ew \cdot \sin t} \cdot \sqrt{ew \cdot \sin t}} \]

      4. rem-square-sqrt 45.2

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

   9. Applied rewrites 45.2%

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

Alternative 3: 88.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{\tan t}\\ t_2 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -1.6 \cdot 10^{+46} \lor \neg \left(eh \leq 4.1 \cdot 10^{+71}\right):\\ \;\;\;\;\left|t\_2 \cdot \sin \tan^{-1} \left(\frac{t\_2}{ew \cdot \sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, ew, \left(t\_1 \cdot \frac{eh}{ew}\right) \cdot \cos t\right)}{\cosh \sinh^{-1} \left(\frac{t\_1}{ew}\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (tan t))) (t_2 (* eh (cos t))))
   (if (or (<= eh -1.6e+46) (not (<= eh 4.1e+71)))
     (fabs (* t_2 (sin (atan (/ t_2 (* ew (sin t)))))))
     (fabs
      (/
       (fma (sin t) ew (* (* t_1 (/ eh ew)) (cos t)))
       (cosh (asinh (/ t_1 ew))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / tan(t);
	double t_2 = eh * cos(t);
	double tmp;
	if ((eh <= -1.6e+46) || !(eh <= 4.1e+71)) {
		tmp = fabs((t_2 * sin(atan((t_2 / (ew * sin(t)))))));
	} else {
		tmp = fabs((fma(sin(t), ew, ((t_1 * (eh / ew)) * cos(t))) / cosh(asinh((t_1 / ew)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / tan(t))
	t_2 = Float64(eh * cos(t))
	tmp = 0.0
	if ((eh <= -1.6e+46) || !(eh <= 4.1e+71))
		tmp = abs(Float64(t_2 * sin(atan(Float64(t_2 / Float64(ew * sin(t)))))));
	else
		tmp = abs(Float64(fma(sin(t), ew, Float64(Float64(t_1 * Float64(eh / ew)) * cos(t))) / cosh(asinh(Float64(t_1 / ew)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -1.6e+46], N[Not[LessEqual[eh, 4.1e+71]], $MachinePrecision]], N[Abs[N[(t$95$2 * N[Sin[N[ArcTan[N[(t$95$2 / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew + N[(N[(t$95$1 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$1 / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.5999999999999999e46 or 4.1000000000000002e71 < eh

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]
   6. 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. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 90.8

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

   7. Applied rewrites 90.8%

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

   if -1.5999999999999999e46 < eh < 4.1000000000000002e71

   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 91.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.2

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

   6. Applied rewrites 94.2%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.6 \cdot 10^{+46} \lor \neg \left(eh \leq 4.1 \cdot 10^{+71}\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, ew, \left(\frac{eh}{\tan t} \cdot \frac{eh}{ew}\right) \cdot \cos t\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{\tan t}\\ t_2 := eh \cdot \cos t\\ t_3 := \left|t\_2 \cdot \sin \tan^{-1} \left(\frac{t\_2}{ew \cdot \sin t}\right)\right|\\ t_4 := \frac{t\_1}{ew}\\ \mathbf{if}\;eh \leq -1.6 \cdot 10^{+46}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq -2.05 \cdot 10^{-111}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, ew, \frac{eh \cdot eh}{\tan t \cdot ew} \cdot \cos t\right)}{\cosh \sinh^{-1} t\_4}\right|\\ \mathbf{elif}\;eh \leq 4.1 \cdot 10^{+71}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, ew, \left(t\_1 \cdot \frac{eh}{ew}\right) \cdot \cos t\right)}{\sqrt{1 + {t\_4}^{2}}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (tan t)))
        (t_2 (* eh (cos t)))
        (t_3 (fabs (* t_2 (sin (atan (/ t_2 (* ew (sin t))))))))
        (t_4 (/ t_1 ew)))
   (if (<= eh -1.6e+46)
     t_3
     (if (<= eh -2.05e-111)
       (fabs
        (/
         (fma (sin t) ew (* (/ (* eh eh) (* (tan t) ew)) (cos t)))
         (cosh (asinh t_4))))
       (if (<= eh 4.1e+71)
         (fabs
          (/
           (fma (sin t) ew (* (* t_1 (/ eh ew)) (cos t)))
           (sqrt (+ 1.0 (pow t_4 2.0)))))
         t_3)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / tan(t);
	double t_2 = eh * cos(t);
	double t_3 = fabs((t_2 * sin(atan((t_2 / (ew * sin(t)))))));
	double t_4 = t_1 / ew;
	double tmp;
	if (eh <= -1.6e+46) {
		tmp = t_3;
	} else if (eh <= -2.05e-111) {
		tmp = fabs((fma(sin(t), ew, (((eh * eh) / (tan(t) * ew)) * cos(t))) / cosh(asinh(t_4))));
	} else if (eh <= 4.1e+71) {
		tmp = fabs((fma(sin(t), ew, ((t_1 * (eh / ew)) * cos(t))) / sqrt((1.0 + pow(t_4, 2.0)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / tan(t))
	t_2 = Float64(eh * cos(t))
	t_3 = abs(Float64(t_2 * sin(atan(Float64(t_2 / Float64(ew * sin(t)))))))
	t_4 = Float64(t_1 / ew)
	tmp = 0.0
	if (eh <= -1.6e+46)
		tmp = t_3;
	elseif (eh <= -2.05e-111)
		tmp = abs(Float64(fma(sin(t), ew, Float64(Float64(Float64(eh * eh) / Float64(tan(t) * ew)) * cos(t))) / cosh(asinh(t_4))));
	elseif (eh <= 4.1e+71)
		tmp = abs(Float64(fma(sin(t), ew, Float64(Float64(t_1 * Float64(eh / ew)) * cos(t))) / sqrt(Float64(1.0 + (t_4 ^ 2.0)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(t$95$2 * N[Sin[N[ArcTan[N[(t$95$2 / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / ew), $MachinePrecision]}, If[LessEqual[eh, -1.6e+46], t$95$3, If[LessEqual[eh, -2.05e-111], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew + N[(N[(N[(eh * eh), $MachinePrecision] / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 4.1e+71], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew + N[(N[(t$95$1 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.5999999999999999e46 or 4.1000000000000002e71 < eh

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]
   6. 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. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 90.8

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

   7. Applied rewrites 90.8%

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

   if -1.5999999999999999e46 < eh < -2.04999999999999984e-111

   1. Initial program 99.7%

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

   4. Applied rewrites 85.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 84.5

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

   6. Applied rewrites 84.5%

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

   if -2.04999999999999984e-111 < eh < 4.1000000000000002e71

   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 92.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.7

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

   6. Applied rewrites 96.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. rem-square-sqrt N/A

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

      5. cosh-asinh N/A

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

      6. lift-asinh.f64 N/A

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

      7. lift-cosh.f64 N/A

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

      8. cosh-asinh N/A

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

      9. lift-asinh.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

   8. Applied rewrites 92.8%

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

Alternative 5: 83.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{\tan t}\\ t_2 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -25000 \lor \neg \left(eh \leq 4.1 \cdot 10^{+71}\right):\\ \;\;\;\;\left|t\_2 \cdot \sin \tan^{-1} \left(\frac{t\_2}{ew \cdot \sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, ew, \left(t\_1 \cdot \frac{eh}{ew}\right) \cdot \cos t\right)}{\sqrt{1 + {\left(\frac{t\_1}{ew}\right)}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (tan t))) (t_2 (* eh (cos t))))
   (if (or (<= eh -25000.0) (not (<= eh 4.1e+71)))
     (fabs (* t_2 (sin (atan (/ t_2 (* ew (sin t)))))))
     (fabs
      (/
       (fma (sin t) ew (* (* t_1 (/ eh ew)) (cos t)))
       (sqrt (+ 1.0 (pow (/ t_1 ew) 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / tan(t);
	double t_2 = eh * cos(t);
	double tmp;
	if ((eh <= -25000.0) || !(eh <= 4.1e+71)) {
		tmp = fabs((t_2 * sin(atan((t_2 / (ew * sin(t)))))));
	} else {
		tmp = fabs((fma(sin(t), ew, ((t_1 * (eh / ew)) * cos(t))) / sqrt((1.0 + pow((t_1 / ew), 2.0)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / tan(t))
	t_2 = Float64(eh * cos(t))
	tmp = 0.0
	if ((eh <= -25000.0) || !(eh <= 4.1e+71))
		tmp = abs(Float64(t_2 * sin(atan(Float64(t_2 / Float64(ew * sin(t)))))));
	else
		tmp = abs(Float64(fma(sin(t), ew, Float64(Float64(t_1 * Float64(eh / ew)) * cos(t))) / sqrt(Float64(1.0 + (Float64(t_1 / ew) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -25000.0], N[Not[LessEqual[eh, 4.1e+71]], $MachinePrecision]], N[Abs[N[(t$95$2 * N[Sin[N[ArcTan[N[(t$95$2 / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew + N[(N[(t$95$1 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(t$95$1 / ew), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -25000 or 4.1000000000000002e71 < eh

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]
   6. 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. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if -25000 < eh < 4.1000000000000002e71

   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%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.6

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

   6. Applied rewrites 95.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. rem-square-sqrt N/A

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

      5. cosh-asinh N/A

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

      6. lift-asinh.f64 N/A

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

      7. lift-cosh.f64 N/A

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

      8. cosh-asinh N/A

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

      9. lift-asinh.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

   8. Applied rewrites 88.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -25000 \lor \neg \left(eh \leq 4.1 \cdot 10^{+71}\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, ew, \left(\frac{eh}{\tan t} \cdot \frac{eh}{ew}\right) \cdot \cos t\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -4.1 \cdot 10^{-111} \lor \neg \left(eh \leq 1.12 \cdot 10^{-41}\right):\\ \;\;\;\;\left|t\_2 \cdot \sin \tan^{-1} \left(\frac{t\_2}{t\_1}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))) (t_2 (* eh (cos t))))
   (if (or (<= eh -4.1e-111) (not (<= eh 1.12e-41)))
     (fabs (* t_2 (sin (atan (/ t_2 t_1)))))
     (fabs t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = eh * cos(t);
	double tmp;
	if ((eh <= -4.1e-111) || !(eh <= 1.12e-41)) {
		tmp = fabs((t_2 * sin(atan((t_2 / t_1)))));
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

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
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * sin(t)
    t_2 = eh * cos(t)
    if ((eh <= (-4.1d-111)) .or. (.not. (eh <= 1.12d-41))) then
        tmp = abs((t_2 * sin(atan((t_2 / t_1)))))
    else
        tmp = abs(t_1)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double t_2 = eh * Math.cos(t);
	double tmp;
	if ((eh <= -4.1e-111) || !(eh <= 1.12e-41)) {
		tmp = Math.abs((t_2 * Math.sin(Math.atan((t_2 / t_1)))));
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	t_2 = eh * math.cos(t)
	tmp = 0
	if (eh <= -4.1e-111) or not (eh <= 1.12e-41):
		tmp = math.fabs((t_2 * math.sin(math.atan((t_2 / t_1)))))
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = Float64(eh * cos(t))
	tmp = 0.0
	if ((eh <= -4.1e-111) || !(eh <= 1.12e-41))
		tmp = abs(Float64(t_2 * sin(atan(Float64(t_2 / t_1)))));
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	t_2 = eh * cos(t);
	tmp = 0.0;
	if ((eh <= -4.1e-111) || ~((eh <= 1.12e-41)))
		tmp = abs((t_2 * sin(atan((t_2 / t_1)))));
	else
		tmp = abs(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -4.1e-111], N[Not[LessEqual[eh, 1.12e-41]], $MachinePrecision]], N[Abs[N[(t$95$2 * N[Sin[N[ArcTan[N[(t$95$2 / t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.09999999999999968e-111 or 1.11999999999999999e-41 < eh

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]
   6. 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. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 80.6

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

   7. Applied rewrites 80.6%

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

   if -4.09999999999999968e-111 < eh < 1.11999999999999999e-41

   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 93.0%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 80.0

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

   7. Applied rewrites 80.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.1 \cdot 10^{-111} \lor \neg \left(eh \leq 1.12 \cdot 10^{-41}\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{\tan t}\\ \mathbf{if}\;t \leq -3.9 \cdot 10^{-56} \lor \neg \left(t \leq 1.55 \cdot 10^{-75}\right):\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, ew, \left(t\_1 \cdot \frac{eh}{ew}\right) \cdot \cos t\right)}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{t\_1}{ew}\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (tan t))))
   (if (or (<= t -3.9e-56) (not (<= t 1.55e-75)))
     (fabs (/ (fma (sin t) ew (* (* t_1 (/ eh ew)) (cos t))) 1.0))
     (fabs (* (tanh (asinh (/ t_1 ew))) eh)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / tan(t);
	double tmp;
	if ((t <= -3.9e-56) || !(t <= 1.55e-75)) {
		tmp = fabs((fma(sin(t), ew, ((t_1 * (eh / ew)) * cos(t))) / 1.0));
	} else {
		tmp = fabs((tanh(asinh((t_1 / ew))) * eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / tan(t))
	tmp = 0.0
	if ((t <= -3.9e-56) || !(t <= 1.55e-75))
		tmp = abs(Float64(fma(sin(t), ew, Float64(Float64(t_1 * Float64(eh / ew)) * cos(t))) / 1.0));
	else
		tmp = abs(Float64(tanh(asinh(Float64(t_1 / ew))) * eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -3.9e-56], N[Not[LessEqual[t, 1.55e-75]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew + N[(N[(t$95$1 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Tanh[N[ArcSinh[N[(t$95$1 / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.9e-56 or 1.55000000000000003e-75 < t

   1. Initial program 99.7%

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

   4. Applied rewrites 71.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 72.3

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

   6. Applied rewrites 72.3%

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

   7. Taylor expanded in eh around 0

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

   9. Applied rewrites 53.0%

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

   if -3.9e-56 < t < 1.55000000000000003e-75

   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 77.4

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

   5. Applied rewrites 77.4%

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

   7. Applied rewrites 77.4%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-56} \lor \neg \left(t \leq 1.55 \cdot 10^{-75}\right):\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, ew, \left(\frac{eh}{\tan t} \cdot \frac{eh}{ew}\right) \cdot \cos t\right)}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2.3 \cdot 10^{-109} \lor \neg \left(eh \leq 4.1 \cdot 10^{+46}\right):\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -2.3e-109) (not (<= eh 4.1e+46)))
   (fabs (* (tanh (asinh (/ (/ eh (tan t)) ew))) eh))
   (fabs (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.3e-109) || !(eh <= 4.1e+46)) {
		tmp = fabs((tanh(asinh(((eh / tan(t)) / ew))) * eh));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -2.3e-109) or not (eh <= 4.1e+46):
		tmp = math.fabs((math.tanh(math.asinh(((eh / math.tan(t)) / ew))) * eh))
	else:
		tmp = math.fabs((ew * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -2.3e-109) || !(eh <= 4.1e+46))
		tmp = abs(Float64(tanh(asinh(Float64(Float64(eh / tan(t)) / ew))) * eh));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -2.3e-109) || ~((eh <= 4.1e+46)))
		tmp = abs((tanh(asinh(((eh / tan(t)) / ew))) * eh));
	else
		tmp = abs((ew * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -2.3e-109], N[Not[LessEqual[eh, 4.1e+46]], $MachinePrecision]], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.3000000000000001e-109 or 4.1e46 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)}\right| \]
   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 55.1

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

   5. Applied rewrites 55.1%

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

   7. Applied rewrites 55.1%

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

   if -2.3000000000000001e-109 < eh < 4.1e46

   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 93.5%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 73.9

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

   7. Applied rewrites 73.9%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.3 \cdot 10^{-109} \lor \neg \left(eh \leq 4.1 \cdot 10^{+46}\right):\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-56} \lor \neg \left(t \leq 1.08 \cdot 10^{-75}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{eh}{ew} \cdot -0.3333333333333333, \frac{eh}{ew}\right)}{t}\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -1.9e-56) (not (<= t 1.08e-75)))
   (fabs (* ew (sin t)))
   (fabs
    (*
     (sin
      (atan (/ (fma (* t t) (* (/ eh ew) -0.3333333333333333) (/ eh ew)) t)))
     eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -1.9e-56) || !(t <= 1.08e-75)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs((sin(atan((fma((t * t), ((eh / ew) * -0.3333333333333333), (eh / ew)) / t))) * eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -1.9e-56) || !(t <= 1.08e-75))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(sin(atan(Float64(fma(Float64(t * t), Float64(Float64(eh / ew) * -0.3333333333333333), Float64(eh / ew)) / t))) * eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -1.9e-56], N[Not[LessEqual[t, 1.08e-75]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(t * t), $MachinePrecision] * N[(N[(eh / ew), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.9000000000000001e-56 or 1.08e-75 < t

   1. Initial program 99.7%

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

   4. Applied rewrites 71.7%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 52.2

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

   7. Applied rewrites 52.2%

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

   if -1.9000000000000001e-56 < t < 1.08e-75

   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 77.4

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

   5. Applied rewrites 77.4%

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

   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 60.2%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-56} \lor \neg \left(t \leq 1.08 \cdot 10^{-75}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{eh}{ew} \cdot -0.3333333333333333, \frac{eh}{ew}\right)}{t}\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \sin t\right| \end{array} \]

FPCore

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

C

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

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((ew * sin(t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Sin[t], $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

4. Applied rewrites 63.0%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 41.5

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

7. Applied rewrites 41.5%

   \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
8. Add Preprocessing

Alternative 11: 17.7% accurate, 36.3× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(-0.16666666666666666 \cdot ew, t \cdot t, ew\right) \cdot t\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (* (fma (* -0.16666666666666666 ew) (* t t) ew) t)))

C

double code(double eh, double ew, double t) {
	return fabs((fma((-0.16666666666666666 * ew), (t * t), ew) * t));
}

Julia

function code(eh, ew, t)
	return abs(Float64(fma(Float64(-0.16666666666666666 * ew), Float64(t * t), ew) * t))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(-0.16666666666666666 * ew), $MachinePrecision] * N[(t * t), $MachinePrecision] + ew), $MachinePrecision] * t), $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

4. Applied rewrites 63.0%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 41.5

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

7. Applied rewrites 41.5%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 18.1%

    \[\leadsto \left|\mathsf{fma}\left(-0.16666666666666666 \cdot ew, t \cdot t, ew\right) \cdot \color{blue}{t}\right| \]
11. Add Preprocessing

Alternative 12: 18.0% accurate, 108.8× speedup

Math

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

FPCore

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

C

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

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((ew * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * t), $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

4. Applied rewrites 63.0%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 41.5

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

7. Applied rewrites 41.5%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 18.0%

    \[\leadsto \left|ew \cdot \color{blue}{t}\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024362 
(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))))))))