Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 9.5s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-neg.f64 99.8

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

8. Applied rewrites 99.8%

   \[\leadsto \left|\mathsf{fma}\left(-eh, \tanh \sinh^{-1} \left(-\frac{eh}{ew} \cdot \tan t\right) \cdot \sin t, \left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(-\frac{eh}{ew} \cdot \tan t\right)}^{2}}}\right)\right| \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-pow.f64 N/A

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

10. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = (-N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision])}, N[Abs[N[((-eh) * N[(N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-neg.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 3: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eh around 0

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

   1. times-frac N/A

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

   2. tan-quot N/A

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

   3. lower-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-*.f64 99.2

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

7. Applied rewrites 99.2%

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

Alternative 4: 99.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[((-eh) * N[(N[Tanh[N[(-1.0 * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[Power[(-t$95$1), 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-neg.f64 99.8

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in eh around 0

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

    1. times-frac N/A

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

    2. lift-/.f64 N/A

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

    3. tan-quot N/A

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

    4. lift-tan.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 99.2

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

11. Applied rewrites 99.2%

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

Alternative 5: 98.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[((-eh) * N[(N[Tanh[N[(-1.0 * N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[Power[(-N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-neg.f64 99.8

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in t around 0

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 98.8

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

11. Applied rewrites 98.8%

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

Alternative 6: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in eh around 0

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 98.5

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

9. Applied rewrites 98.5%

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

Alternative 7: 88.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-1 \cdot \left(eh \cdot t\right)}{ew}\\ t_2 := ew \cdot \cos t\\ t_3 := \left|t\_2 \cdot \frac{1}{\sqrt{1 + t\_1 \cdot t\_1}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \mathbf{if}\;eh \leq -2.3 \cdot 10^{-116}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq 7.5 \cdot 10^{+54}:\\ \;\;\;\;\left|t\_2\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (* -1.0 (* eh t)) ew))
        (t_2 (* ew (cos t)))
        (t_3
         (fabs
          (-
           (* t_2 (/ 1.0 (sqrt (+ 1.0 (* t_1 t_1)))))
           (* (* eh (sin t)) (sin (atan (/ (* (- eh) t) ew))))))))
   (if (<= eh -2.3e-116) t_3 (if (<= eh 7.5e+54) (fabs t_2) t_3))))

C

double code(double eh, double ew, double t) {
	double t_1 = (-1.0 * (eh * t)) / ew;
	double t_2 = ew * cos(t);
	double t_3 = fabs(((t_2 * (1.0 / sqrt((1.0 + (t_1 * t_1))))) - ((eh * sin(t)) * sin(atan(((-eh * t) / ew))))));
	double tmp;
	if (eh <= -2.3e-116) {
		tmp = t_3;
	} else if (eh <= 7.5e+54) {
		tmp = fabs(t_2);
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = ((-1.0d0) * (eh * t)) / ew
    t_2 = ew * cos(t)
    t_3 = abs(((t_2 * (1.0d0 / sqrt((1.0d0 + (t_1 * t_1))))) - ((eh * sin(t)) * sin(atan(((-eh * t) / ew))))))
    if (eh <= (-2.3d-116)) then
        tmp = t_3
    else if (eh <= 7.5d+54) then
        tmp = abs(t_2)
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (-1.0 * (eh * t)) / ew;
	double t_2 = ew * Math.cos(t);
	double t_3 = Math.abs(((t_2 * (1.0 / Math.sqrt((1.0 + (t_1 * t_1))))) - ((eh * Math.sin(t)) * Math.sin(Math.atan(((-eh * t) / ew))))));
	double tmp;
	if (eh <= -2.3e-116) {
		tmp = t_3;
	} else if (eh <= 7.5e+54) {
		tmp = Math.abs(t_2);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (-1.0 * (eh * t)) / ew
	t_2 = ew * math.cos(t)
	t_3 = math.fabs(((t_2 * (1.0 / math.sqrt((1.0 + (t_1 * t_1))))) - ((eh * math.sin(t)) * math.sin(math.atan(((-eh * t) / ew))))))
	tmp = 0
	if eh <= -2.3e-116:
		tmp = t_3
	elif eh <= 7.5e+54:
		tmp = math.fabs(t_2)
	else:
		tmp = t_3
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-1.0 * Float64(eh * t)) / ew)
	t_2 = Float64(ew * cos(t))
	t_3 = abs(Float64(Float64(t_2 * Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_1))))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-eh) * t) / ew))))))
	tmp = 0.0
	if (eh <= -2.3e-116)
		tmp = t_3;
	elseif (eh <= 7.5e+54)
		tmp = abs(t_2);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (-1.0 * (eh * t)) / ew;
	t_2 = ew * cos(t);
	t_3 = abs(((t_2 * (1.0 / sqrt((1.0 + (t_1 * t_1))))) - ((eh * sin(t)) * sin(atan(((-eh * t) / ew))))));
	tmp = 0.0;
	if (eh <= -2.3e-116)
		tmp = t_3;
	elseif (eh <= 7.5e+54)
		tmp = abs(t_2);
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(-1.0 * N[(eh * t), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(N[(t$95$2 * N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.3e-116], t$95$3, If[LessEqual[eh, 7.5e+54], N[Abs[t$95$2], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.30000000000000002e-116 or 7.50000000000000042e54 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 91.5

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

   7. Applied rewrites 91.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 91.3

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

   9. Applied rewrites 91.3%

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

   if -2.30000000000000002e-116 < eh < 7.50000000000000042e54

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in eh around 0

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

      1. lift-cos.f64 N/A

         \[\leadsto \left|ew \cdot \cos t\right| \]

      2. lift-*.f64 83.8

         \[\leadsto \left|ew \cdot \cos t\right| \]

   9. Applied rewrites 83.8%

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

Alternative 8: 74.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -2.3 \cdot 10^{-117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 3.1 \cdot 10^{-124}:\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(-\frac{eh}{ew} \cdot \tan t\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -2.3e-117)
     t_1
     (if (<= ew 3.1e-124)
       (fabs (* (- eh) (* (tanh (asinh (- (* (/ eh ew) (tan t))))) (sin t))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -2.3e-117) {
		tmp = t_1;
	} else if (ew <= 3.1e-124) {
		tmp = fabs((-eh * (tanh(asinh(-((eh / ew) * tan(t)))) * sin(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -2.3e-117:
		tmp = t_1
	elif ew <= 3.1e-124:
		tmp = math.fabs((-eh * (math.tanh(math.asinh(-((eh / ew) * math.tan(t)))) * math.sin(t))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -2.3e-117)
		tmp = t_1;
	elseif (ew <= 3.1e-124)
		tmp = abs(Float64(Float64(-eh) * Float64(tanh(asinh(Float64(-Float64(Float64(eh / ew) * tan(t))))) * sin(t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -2.3e-117)
		tmp = t_1;
	elseif (ew <= 3.1e-124)
		tmp = abs((-eh * (tanh(asinh(-((eh / ew) * tan(t)))) * sin(t))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.3e-117], t$95$1, If[LessEqual[ew, 3.1e-124], N[Abs[N[((-eh) * N[(N[Tanh[N[ArcSinh[(-N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.29999999999999994e-117 or 3.0999999999999998e-124 < ew

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in eh around 0

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

      1. lift-cos.f64 N/A

         \[\leadsto \left|ew \cdot \cos t\right| \]

      2. lift-*.f64 75.5

         \[\leadsto \left|ew \cdot \cos t\right| \]

   9. Applied rewrites 75.5%

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

   if -2.29999999999999994e-117 < ew < 3.0999999999999998e-124

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 73.5%

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

Alternative 9: 61.5% accurate, 8.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((ew * cos(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 * cos(t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in eh around 0

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

   1. lift-cos.f64 N/A

      \[\leadsto \left|ew \cdot \cos t\right| \]

   2. lift-*.f64 61.5

      \[\leadsto \left|ew \cdot \cos t\right| \]

9. Applied rewrites 61.5%

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

Alternative 10: 41.5% accurate, 287.3× speedup

Math

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

FPCore

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

C

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

Fortran

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

   \[\leadsto \left|ew\right| \]
8. Step-by-step derivation

9. Applied rewrites 41.5%

   \[\leadsto \left|ew\right| \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025099 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))