Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 22.9s

Alternatives: 12

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in t around inf

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

   1. lift-sin.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 89.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{t \cdot ew}\\ t_2 := eh \cdot \cos t\\ t_3 := ew \cdot \sin t\\ t_4 := \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-0.3333333333333333 \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right)\\ \mathbf{if}\;ew \leq 4.4 \cdot 10^{-57}:\\ \;\;\;\;\left|t\_3 \cdot \cos t\_4 + t\_2 \cdot \sin t\_4\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(t\_3, \frac{1}{\sqrt{1 + t\_1 \cdot t\_1}}, t\_2 \cdot \tanh \sinh^{-1} t\_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* t ew)))
        (t_2 (* eh (cos t)))
        (t_3 (* ew (sin t)))
        (t_4 (atan (/ (/ (fma (* -0.3333333333333333 (* t t)) eh eh) ew) t))))
   (if (<= ew 4.4e-57)
     (fabs (+ (* t_3 (cos t_4)) (* t_2 (sin t_4))))
     (fabs
      (fma
       t_3
       (/ 1.0 (sqrt (+ 1.0 (* t_1 t_1))))
       (* t_2 (tanh (asinh t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (t * ew);
	double t_2 = eh * cos(t);
	double t_3 = ew * sin(t);
	double t_4 = atan(((fma((-0.3333333333333333 * (t * t)), eh, eh) / ew) / t));
	double tmp;
	if (ew <= 4.4e-57) {
		tmp = fabs(((t_3 * cos(t_4)) + (t_2 * sin(t_4))));
	} else {
		tmp = fabs(fma(t_3, (1.0 / sqrt((1.0 + (t_1 * t_1)))), (t_2 * tanh(asinh(t_1)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(t * ew))
	t_2 = Float64(eh * cos(t))
	t_3 = Float64(ew * sin(t))
	t_4 = atan(Float64(Float64(fma(Float64(-0.3333333333333333 * Float64(t * t)), eh, eh) / ew) / t))
	tmp = 0.0
	if (ew <= 4.4e-57)
		tmp = abs(Float64(Float64(t_3 * cos(t_4)) + Float64(t_2 * sin(t_4))));
	else
		tmp = abs(fma(t_3, Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_1)))), Float64(t_2 * tanh(asinh(t_1)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(N[(N[(N[(-0.3333333333333333 * N[(t * t), $MachinePrecision]), $MachinePrecision] * eh + eh), $MachinePrecision] / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, 4.4e-57], N[Abs[N[(N[(t$95$3 * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$3 * N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if ew < 4.39999999999999997e-57

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\frac{-1}{3} \cdot \left(eh \cdot {t}^{2}\right)}{ew} + \frac{eh}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      3. div-add-rev N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\frac{-1}{3} \cdot \left(eh \cdot {t}^{2}\right) + eh}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\frac{-1}{3} \cdot \left(eh \cdot {t}^{2}\right) + eh}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      5. *-commutative N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\frac{-1}{3} \cdot \left({t}^{2} \cdot eh\right) + eh}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      6. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\left(\frac{-1}{3} \cdot {t}^{2}\right) \cdot eh + eh}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      7. metadata-eval N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot {t}^{2}\right) \cdot eh + eh}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      8. lower-fma.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot {t}^{2}, eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      9. metadata-eval N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot {t}^{2}, eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot {t}^{2}, eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      11. unpow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      12. lower-*.f64 86.6

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

   4. Applied rewrites 86.6%

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

   5. Taylor expanded in t around 0

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}\right| \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{\color{blue}{t}}\right)\right| \]

      2. associate-*r/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\frac{-1}{3} \cdot \left(eh \cdot {t}^{2}\right)}{ew} + \frac{eh}{ew}}{t}\right)\right| \]

      3. div-add-rev N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\frac{-1}{3} \cdot \left(eh \cdot {t}^{2}\right) + eh}{ew}}{t}\right)\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\frac{-1}{3} \cdot \left(eh \cdot {t}^{2}\right) + eh}{ew}}{t}\right)\right| \]

      5. *-commutative N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\frac{-1}{3} \cdot \left({t}^{2} \cdot eh\right) + eh}{ew}}{t}\right)\right| \]

      6. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\left(\frac{-1}{3} \cdot {t}^{2}\right) \cdot eh + eh}{ew}}{t}\right)\right| \]

      7. metadata-eval N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot {t}^{2}\right) \cdot eh + eh}{ew}}{t}\right)\right| \]

      8. lower-fma.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot {t}^{2}, eh, eh\right)}{ew}}{t}\right)\right| \]

      9. metadata-eval N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot {t}^{2}, eh, eh\right)}{ew}}{t}\right)\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot {t}^{2}, eh, eh\right)}{ew}}{t}\right)\right| \]

      11. unpow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{3} \cdot \left(t \cdot t\right), eh, eh\right)}{ew}}{t}\right)\right| \]

      12. lower-*.f64 86.6

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

   7. Applied rewrites 86.6%

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

   if 4.39999999999999997e-57 < ew

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.8

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

   7. Applied rewrites 90.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 75.3%

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

       1. lift-/.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. tanh-asinh-rev N/A

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

       6. lower-tanh.f64 N/A

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

   11. Applied rewrites 90.8%

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

Alternative 3: 87.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{t \cdot ew}\\ \mathbf{if}\;ew \leq 10^{-180}:\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\cos t \cdot \frac{eh}{\sin t \cdot ew}\right) \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, \frac{1}{\sqrt{1 + t\_1 \cdot t\_1}}, \left(eh \cdot \cos t\right) \cdot \tanh \sinh^{-1} t\_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* t ew))))
   (if (<= ew 1e-180)
     (fabs (* (* (tanh (asinh (* (cos t) (/ eh (* (sin t) ew))))) (cos t)) eh))
     (fabs
      (fma
       (* ew (sin t))
       (/ 1.0 (sqrt (+ 1.0 (* t_1 t_1))))
       (* (* eh (cos t)) (tanh (asinh t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (t * ew);
	double tmp;
	if (ew <= 1e-180) {
		tmp = fabs(((tanh(asinh((cos(t) * (eh / (sin(t) * ew))))) * cos(t)) * eh));
	} else {
		tmp = fabs(fma((ew * sin(t)), (1.0 / sqrt((1.0 + (t_1 * t_1)))), ((eh * cos(t)) * tanh(asinh(t_1)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(t * ew))
	tmp = 0.0
	if (ew <= 1e-180)
		tmp = abs(Float64(Float64(tanh(asinh(Float64(cos(t) * Float64(eh / Float64(sin(t) * ew))))) * cos(t)) * eh));
	else
		tmp = abs(fma(Float64(ew * sin(t)), Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_1)))), Float64(Float64(eh * cos(t)) * tanh(asinh(t_1)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, 1e-180], N[Abs[N[(N[(N[Tanh[N[ArcSinh[N[(N[Cos[t], $MachinePrecision] * N[(eh / N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < 1e-180

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 66.3%

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

   if 1e-180 < ew

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.4

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

   7. Applied rewrites 90.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 66.0%

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

       1. lift-/.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. tanh-asinh-rev N/A

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

       6. lower-tanh.f64 N/A

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

   11. Applied rewrites 90.4%

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

Alternative 4: 76.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 89.7

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

7. Applied rewrites 89.7%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lower-fma.f64 N/A

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

9. Applied rewrites 57.5%

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

    1. lift-/.f64 N/A

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

    2. lift-sqrt.f64 N/A

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

    3. lift-+.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. tanh-asinh-rev N/A

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

    6. lower-tanh.f64 N/A

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

11. Applied rewrites 89.7%

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

Alternative 5: 74.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{t \cdot ew}\\ t_2 := eh \cdot \cos t\\ t_3 := \sqrt{1 + t\_1 \cdot t\_1}\\ \mathbf{if}\;t \leq 1600:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot t, \frac{1}{t\_3}, t\_2 \cdot \tanh \sinh^{-1} t\_1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, 1, t\_2 \cdot \frac{t\_1}{t\_3}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* t ew)))
        (t_2 (* eh (cos t)))
        (t_3 (sqrt (+ 1.0 (* t_1 t_1)))))
   (if (<= t 1600.0)
     (fabs (fma (* ew t) (/ 1.0 t_3) (* t_2 (tanh (asinh t_1)))))
     (fabs (fma (* ew (sin t)) 1.0 (* t_2 (/ t_1 t_3)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (t * ew);
	double t_2 = eh * cos(t);
	double t_3 = sqrt((1.0 + (t_1 * t_1)));
	double tmp;
	if (t <= 1600.0) {
		tmp = fabs(fma((ew * t), (1.0 / t_3), (t_2 * tanh(asinh(t_1)))));
	} else {
		tmp = fabs(fma((ew * sin(t)), 1.0, (t_2 * (t_1 / t_3))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(t * ew))
	t_2 = Float64(eh * cos(t))
	t_3 = sqrt(Float64(1.0 + Float64(t_1 * t_1)))
	tmp = 0.0
	if (t <= 1600.0)
		tmp = abs(fma(Float64(ew * t), Float64(1.0 / t_3), Float64(t_2 * tanh(asinh(t_1)))));
	else
		tmp = abs(fma(Float64(ew * sin(t)), 1.0, Float64(t_2 * Float64(t_1 / t_3))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1600.0], N[Abs[N[(N[(ew * t), $MachinePrecision] * N[(1.0 / t$95$3), $MachinePrecision] + N[(t$95$2 * N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * 1.0 + N[(t$95$2 * N[(t$95$1 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if t < 1600

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 93.1

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

   7. Applied rewrites 93.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 54.9%

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

       1. lift-/.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. tanh-asinh-rev N/A

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

       6. lower-tanh.f64 N/A

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

   11. Applied rewrites 93.1%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 77.2%

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

   if 1600 < t

   1. Initial program 99.6%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 79.0

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

   7. Applied rewrites 79.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 65.6%

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

   10. Taylor expanded in eh around 0

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

   12. Applied rewrites 65.7%

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

Alternative 6: 63.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{t \cdot ew}\\ t_2 := \sqrt{1 + t\_1 \cdot t\_1}\\ t_3 := \frac{1}{t\_2}\\ \mathbf{if}\;eh \leq 9.5 \cdot 10^{-34}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, t\_3, eh \cdot \frac{t\_1}{t\_2}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot t, t\_3, \left(eh \cdot \cos t\right) \cdot \tanh \sinh^{-1} t\_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* t ew)))
        (t_2 (sqrt (+ 1.0 (* t_1 t_1))))
        (t_3 (/ 1.0 t_2)))
   (if (<= eh 9.5e-34)
     (fabs (fma (* ew (sin t)) t_3 (* eh (/ t_1 t_2))))
     (fabs (fma (* ew t) t_3 (* (* eh (cos t)) (tanh (asinh t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (t * ew);
	double t_2 = sqrt((1.0 + (t_1 * t_1)));
	double t_3 = 1.0 / t_2;
	double tmp;
	if (eh <= 9.5e-34) {
		tmp = fabs(fma((ew * sin(t)), t_3, (eh * (t_1 / t_2))));
	} else {
		tmp = fabs(fma((ew * t), t_3, ((eh * cos(t)) * tanh(asinh(t_1)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(t * ew))
	t_2 = sqrt(Float64(1.0 + Float64(t_1 * t_1)))
	t_3 = Float64(1.0 / t_2)
	tmp = 0.0
	if (eh <= 9.5e-34)
		tmp = abs(fma(Float64(ew * sin(t)), t_3, Float64(eh * Float64(t_1 / t_2))));
	else
		tmp = abs(fma(Float64(ew * t), t_3, Float64(Float64(eh * cos(t)) * tanh(asinh(t_1)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / t$95$2), $MachinePrecision]}, If[LessEqual[eh, 9.5e-34], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$3 + N[(eh * N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * t), $MachinePrecision] * t$95$3 + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < 9.49999999999999985e-34

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

   7. Applied rewrites 90.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 64.2%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 59.6%

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

   if 9.49999999999999985e-34 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.7

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

   7. Applied rewrites 87.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 39.9%

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

       1. lift-/.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. tanh-asinh-rev N/A

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

       6. lower-tanh.f64 N/A

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

   11. Applied rewrites 87.7%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 74.7%

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

Alternative 7: 57.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{t \cdot ew}\\ \mathbf{if}\;eh \leq 1.68 \cdot 10^{-106}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot t, \frac{1}{\sqrt{1 + t\_1 \cdot t\_1}}, \left(eh \cdot \cos t\right) \cdot \tanh \sinh^{-1} t\_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* t ew))))
   (if (<= eh 1.68e-106)
     (fabs (* ew (sin t)))
     (fabs
      (fma
       (* ew t)
       (/ 1.0 (sqrt (+ 1.0 (* t_1 t_1))))
       (* (* eh (cos t)) (tanh (asinh t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (t * ew);
	double tmp;
	if (eh <= 1.68e-106) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs(fma((ew * t), (1.0 / sqrt((1.0 + (t_1 * t_1)))), ((eh * cos(t)) * tanh(asinh(t_1)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(t * ew))
	tmp = 0.0
	if (eh <= 1.68e-106)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(fma(Float64(ew * t), Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_1)))), Float64(Float64(eh * cos(t)) * tanh(asinh(t_1)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, 1.68e-106], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * t), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < 1.6799999999999999e-106

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 50.5

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

   6. Applied rewrites 50.5%

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

   if 1.6799999999999999e-106 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.6

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

   7. Applied rewrites 87.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 44.6%

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

       1. lift-/.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. tanh-asinh-rev N/A

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

       6. lower-tanh.f64 N/A

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

   11. Applied rewrites 87.6%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 72.3%

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

Alternative 8: 52.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{t \cdot ew}\\ t_2 := \sqrt{1 + t\_1 \cdot t\_1}\\ \mathbf{if}\;t \leq 7.6 \cdot 10^{-65}:\\ \;\;\;\;\left|\frac{eh}{ew \cdot \frac{1}{ew}}\right|\\ \mathbf{elif}\;t \leq 0.00048:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot t, \frac{1}{t\_2}, \left(eh \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.041666666666666664 \cdot \left(t \cdot t\right) - 0.5\right)\right)\right) \cdot \frac{t\_1}{t\_2}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* t ew))) (t_2 (sqrt (+ 1.0 (* t_1 t_1)))))
   (if (<= t 7.6e-65)
     (fabs (/ eh (* ew (/ 1.0 ew))))
     (if (<= t 0.00048)
       (fabs
        (fma
         (* ew t)
         (/ 1.0 t_2)
         (*
          (* eh (+ 1.0 (* (* t t) (- (* 0.041666666666666664 (* t t)) 0.5))))
          (/ t_1 t_2))))
       (fabs (* ew (sin t)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (t * ew);
	double t_2 = sqrt((1.0 + (t_1 * t_1)));
	double tmp;
	if (t <= 7.6e-65) {
		tmp = fabs((eh / (ew * (1.0 / ew))));
	} else if (t <= 0.00048) {
		tmp = fabs(fma((ew * t), (1.0 / t_2), ((eh * (1.0 + ((t * t) * ((0.041666666666666664 * (t * t)) - 0.5)))) * (t_1 / t_2))));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(t * ew))
	t_2 = sqrt(Float64(1.0 + Float64(t_1 * t_1)))
	tmp = 0.0
	if (t <= 7.6e-65)
		tmp = abs(Float64(eh / Float64(ew * Float64(1.0 / ew))));
	elseif (t <= 0.00048)
		tmp = abs(fma(Float64(ew * t), Float64(1.0 / t_2), Float64(Float64(eh * Float64(1.0 + Float64(Float64(t * t) * Float64(Float64(0.041666666666666664 * Float64(t * t)) - 0.5)))) * Float64(t_1 / t_2))));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 7.6e-65], N[Abs[N[(eh / N[(ew * N[(1.0 / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 0.00048], N[Abs[N[(N[(ew * t), $MachinePrecision] * N[(1.0 / t$95$2), $MachinePrecision] + N[(N[(eh * N[(1.0 + N[(N[(t * t), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(t * t), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.6000000000000003e-65

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 92.6

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

   7. Applied rewrites 92.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 53.7%

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

   10. Taylor expanded in t around 0

       \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   11. Step-by-step derivation

       1. cos-atan-rev N/A

          \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       2. sin-atan-rev N/A

          \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       3. lower-/.f64 N/A

          \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

       4. unpow2 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       5. lower-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       6. lower-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   12. Applied rewrites 13.6%

       \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   13. Taylor expanded in eh around 0

       \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}}\right| \]
   14. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{1}{{ew}^{2}}}}}\right| \]

       2. pow-flip N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{{ew}^{\left(\mathsf{neg}\left(2\right)\right)}}}\right| \]

       3. metadata-eval N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{{ew}^{-2}}}\right| \]

       4. sqrt-pow1 N/A

          \[\leadsto \left|\frac{eh}{ew \cdot {ew}^{\left(\frac{-2}{\color{blue}{2}}\right)}}\right| \]

       5. metadata-eval N/A

          \[\leadsto \left|\frac{eh}{ew \cdot {ew}^{-1}}\right| \]

       6. inv-pow N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \frac{1}{ew}}\right| \]

       7. lower-*.f64 N/A

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

       8. lower-/.f64 51.4

          \[\leadsto \left|\frac{eh}{ew \cdot \frac{1}{ew}}\right| \]

   15. Applied rewrites 51.4%

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

   if 7.6000000000000003e-65 < t < 4.80000000000000012e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 68.9%

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

   10. Taylor expanded in t around 0

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

       1. lift-*.f64 68.1

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

   12. Applied rewrites 68.1%

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

   13. Taylor expanded in t around 0

       \[\leadsto \left|\mathsf{fma}\left(ew \cdot t, \frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}, \left(eh \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{24} \cdot {t}^{2} - \frac{1}{2}\right)\right)}\right) \cdot \frac{\frac{eh}{t \cdot ew}}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}\right)\right| \]
   14. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew \cdot t, \frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}, \left(eh \cdot \left(1 + \color{blue}{{t}^{2} \cdot \left(\frac{1}{24} \cdot {t}^{2} - \frac{1}{2}\right)}\right)\right) \cdot \frac{\frac{eh}{t \cdot ew}}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}\right)\right| \]

       2. lower-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew \cdot t, \frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}, \left(eh \cdot \left(1 + {t}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {t}^{2} - \frac{1}{2}\right)}\right)\right) \cdot \frac{\frac{eh}{t \cdot ew}}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}\right)\right| \]

       3. unpow2 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew \cdot t, \frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}, \left(eh \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {t}^{2}} - \frac{1}{2}\right)\right)\right) \cdot \frac{\frac{eh}{t \cdot ew}}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}\right)\right| \]

       4. lower-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew \cdot t, \frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}, \left(eh \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {t}^{2}} - \frac{1}{2}\right)\right)\right) \cdot \frac{\frac{eh}{t \cdot ew}}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}\right)\right| \]

       5. lower--.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew \cdot t, \frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}, \left(eh \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{24} \cdot {t}^{2} - \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \frac{\frac{eh}{t \cdot ew}}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}\right)\right| \]

       6. lower-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew \cdot t, \frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}, \left(eh \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{24} \cdot {t}^{2} - \frac{1}{2}\right)\right)\right) \cdot \frac{\frac{eh}{t \cdot ew}}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}\right)\right| \]

       7. unpow2 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew \cdot t, \frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}, \left(eh \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{24} \cdot \left(t \cdot t\right) - \frac{1}{2}\right)\right)\right) \cdot \frac{\frac{eh}{t \cdot ew}}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}\right)\right| \]

       8. lower-*.f64 68.1

          \[\leadsto \left|\mathsf{fma}\left(ew \cdot t, \frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}, \left(eh \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.041666666666666664 \cdot \left(t \cdot t\right) - 0.5\right)\right)\right) \cdot \frac{\frac{eh}{t \cdot ew}}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}\right)\right| \]

   15. Applied rewrites 68.1%

       \[\leadsto \left|\mathsf{fma}\left(ew \cdot t, \frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}, \left(eh \cdot \color{blue}{\left(1 + \left(t \cdot t\right) \cdot \left(0.041666666666666664 \cdot \left(t \cdot t\right) - 0.5\right)\right)}\right) \cdot \frac{\frac{eh}{t \cdot ew}}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}\right)\right| \]

   if 4.80000000000000012e-4 < t

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

Alternative 9: 52.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{t \cdot ew}\\ t_2 := \sqrt{1 + t\_1 \cdot t\_1}\\ \mathbf{if}\;t \leq 7.6 \cdot 10^{-65}:\\ \;\;\;\;\left|\frac{eh}{ew \cdot \frac{1}{ew}}\right|\\ \mathbf{elif}\;t \leq 0.00035:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot t, \frac{1}{t\_2}, \left(eh \cdot \left(1 - 0.5 \cdot \left(t \cdot t\right)\right)\right) \cdot \frac{t\_1}{t\_2}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* t ew))) (t_2 (sqrt (+ 1.0 (* t_1 t_1)))))
   (if (<= t 7.6e-65)
     (fabs (/ eh (* ew (/ 1.0 ew))))
     (if (<= t 0.00035)
       (fabs
        (fma
         (* ew t)
         (/ 1.0 t_2)
         (* (* eh (- 1.0 (* 0.5 (* t t)))) (/ t_1 t_2))))
       (fabs (* ew (sin t)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (t * ew);
	double t_2 = sqrt((1.0 + (t_1 * t_1)));
	double tmp;
	if (t <= 7.6e-65) {
		tmp = fabs((eh / (ew * (1.0 / ew))));
	} else if (t <= 0.00035) {
		tmp = fabs(fma((ew * t), (1.0 / t_2), ((eh * (1.0 - (0.5 * (t * t)))) * (t_1 / t_2))));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(t * ew))
	t_2 = sqrt(Float64(1.0 + Float64(t_1 * t_1)))
	tmp = 0.0
	if (t <= 7.6e-65)
		tmp = abs(Float64(eh / Float64(ew * Float64(1.0 / ew))));
	elseif (t <= 0.00035)
		tmp = abs(fma(Float64(ew * t), Float64(1.0 / t_2), Float64(Float64(eh * Float64(1.0 - Float64(0.5 * Float64(t * t)))) * Float64(t_1 / t_2))));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 7.6e-65], N[Abs[N[(eh / N[(ew * N[(1.0 / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 0.00035], N[Abs[N[(N[(ew * t), $MachinePrecision] * N[(1.0 / t$95$2), $MachinePrecision] + N[(N[(eh * N[(1.0 - N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.6000000000000003e-65

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 92.6

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

   7. Applied rewrites 92.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 53.7%

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

   10. Taylor expanded in t around 0

       \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   11. Step-by-step derivation

       1. cos-atan-rev N/A

          \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       2. sin-atan-rev N/A

          \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       3. lower-/.f64 N/A

          \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

       4. unpow2 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       5. lower-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       6. lower-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   12. Applied rewrites 13.6%

       \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   13. Taylor expanded in eh around 0

       \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}}\right| \]
   14. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{1}{{ew}^{2}}}}}\right| \]

       2. pow-flip N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{{ew}^{\left(\mathsf{neg}\left(2\right)\right)}}}\right| \]

       3. metadata-eval N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{{ew}^{-2}}}\right| \]

       4. sqrt-pow1 N/A

          \[\leadsto \left|\frac{eh}{ew \cdot {ew}^{\left(\frac{-2}{\color{blue}{2}}\right)}}\right| \]

       5. metadata-eval N/A

          \[\leadsto \left|\frac{eh}{ew \cdot {ew}^{-1}}\right| \]

       6. inv-pow N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \frac{1}{ew}}\right| \]

       7. lower-*.f64 N/A

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

       8. lower-/.f64 51.4

          \[\leadsto \left|\frac{eh}{ew \cdot \frac{1}{ew}}\right| \]

   15. Applied rewrites 51.4%

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

   if 7.6000000000000003e-65 < t < 3.49999999999999996e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 68.9%

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

   10. Taylor expanded in t around 0

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

       1. lift-*.f64 68.1

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

   12. Applied rewrites 68.1%

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

   13. Taylor expanded in t around 0

       \[\leadsto \left|\mathsf{fma}\left(ew \cdot t, \frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}, \left(eh \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {t}^{2}\right)}\right) \cdot \frac{\frac{eh}{t \cdot ew}}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}\right)\right| \]
   14. Step-by-step derivation

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

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

       2. lower--.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 68.1

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

   15. Applied rewrites 68.1%

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

   if 3.49999999999999996e-4 < t

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

Alternative 10: 51.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.00016:\\ \;\;\;\;\left|\frac{eh}{ew \cdot \frac{1}{ew}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t 0.00016) (fabs (/ eh (* ew (/ 1.0 ew)))) (fabs (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= 0.00016) {
		tmp = fabs((eh / (ew * (1.0 / ew))));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	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) :: tmp
    if (t <= 0.00016d0) then
        tmp = abs((eh / (ew * (1.0d0 / ew))))
    else
        tmp = abs((ew * sin(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (t <= 0.00016) {
		tmp = Math.abs((eh / (ew * (1.0 / ew))));
	} else {
		tmp = Math.abs((ew * Math.sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if t <= 0.00016:
		tmp = math.fabs((eh / (ew * (1.0 / ew))))
	else:
		tmp = math.fabs((ew * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= 0.00016)
		tmp = abs(Float64(eh / Float64(ew * Float64(1.0 / ew))));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (t <= 0.00016)
		tmp = abs((eh / (ew * (1.0 / ew))));
	else
		tmp = abs((ew * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, 0.00016], N[Abs[N[(eh / N[(ew * N[(1.0 / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.60000000000000013e-4

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 93.1

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

   7. Applied rewrites 93.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 54.8%

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

   10. Taylor expanded in t around 0

       \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   11. Step-by-step derivation

       1. cos-atan-rev N/A

          \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       2. sin-atan-rev N/A

          \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       3. lower-/.f64 N/A

          \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

       4. unpow2 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       5. lower-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       6. lower-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   12. Applied rewrites 13.4%

       \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   13. Taylor expanded in eh around 0

       \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}}\right| \]
   14. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{1}{{ew}^{2}}}}}\right| \]

       2. pow-flip N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{{ew}^{\left(\mathsf{neg}\left(2\right)\right)}}}\right| \]

       3. metadata-eval N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{{ew}^{-2}}}\right| \]

       4. sqrt-pow1 N/A

          \[\leadsto \left|\frac{eh}{ew \cdot {ew}^{\left(\frac{-2}{\color{blue}{2}}\right)}}\right| \]

       5. metadata-eval N/A

          \[\leadsto \left|\frac{eh}{ew \cdot {ew}^{-1}}\right| \]

       6. inv-pow N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \frac{1}{ew}}\right| \]

       7. lower-*.f64 N/A

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

       8. lower-/.f64 51.5

          \[\leadsto \left|\frac{eh}{ew \cdot \frac{1}{ew}}\right| \]

   15. Applied rewrites 51.5%

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

   if 1.60000000000000013e-4 < t

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

Alternative 11: 42.1% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 4.4 \cdot 10^{-56}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{eh}{ew \cdot \frac{1}{ew}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 4.4e-56) (* ew (sin t)) (fabs (/ eh (* ew (/ 1.0 ew))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 4.4e-56) {
		tmp = ew * sin(t);
	} else {
		tmp = fabs((eh / (ew * (1.0 / ew))));
	}
	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) :: tmp
    if (eh <= 4.4d-56) then
        tmp = ew * sin(t)
    else
        tmp = abs((eh / (ew * (1.0d0 / ew))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 4.4e-56) {
		tmp = ew * Math.sin(t);
	} else {
		tmp = Math.abs((eh / (ew * (1.0 / ew))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= 4.4e-56:
		tmp = ew * math.sin(t)
	else:
		tmp = math.fabs((eh / (ew * (1.0 / ew))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 4.4e-56)
		tmp = Float64(ew * sin(t));
	else
		tmp = abs(Float64(eh / Float64(ew * Float64(1.0 / ew))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= 4.4e-56)
		tmp = ew * sin(t);
	else
		tmp = abs((eh / (ew * (1.0 / ew))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 4.4e-56], N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], N[Abs[N[(eh / N[(ew * N[(1.0 / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 4.40000000000000008e-56

   1. Initial program 99.8%

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

   3. Applied rewrites 49.3%

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

   4. Taylor expanded in ew around -inf

      \[\leadsto \color{blue}{{ew}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{{eh}^{2} \cdot {\cos t}^{2}}{\sin t} + \frac{1}{2} \cdot \frac{{eh}^{2} \cdot {\cos t}^{2}}{\sin t}}{{ew}^{3}} + {\left(\sqrt{\frac{\sin t}{ew}}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \color{blue}{{ew}^{2}} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{{eh}^{2} \cdot {\cos t}^{2}}{\sin t} + \frac{1}{2} \cdot \frac{{eh}^{2} \cdot {\cos t}^{2}}{\sin t}}{{ew}^{3}} + {\left(\sqrt{\frac{\sin t}{ew}}\right)}^{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto {ew}^{2} \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{{eh}^{2} \cdot {\cos t}^{2}}{\sin t} + \frac{1}{2} \cdot \frac{{eh}^{2} \cdot {\cos t}^{2}}{\sin t}}{{ew}^{3}} + {\left(\sqrt{\frac{\sin t}{ew}}\right)}^{2}\right)} \]

   6. Applied rewrites 12.5%

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

   7. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

         \[\leadsto ew \cdot \sin t \]

      2. lift-*.f64 26.1

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

   9. Applied rewrites 26.1%

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

   if 4.40000000000000008e-56 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.7

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

   7. Applied rewrites 87.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 41.5%

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

   10. Taylor expanded in t around 0

       \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   11. Step-by-step derivation

       1. cos-atan-rev N/A

          \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       2. sin-atan-rev N/A

          \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       3. lower-/.f64 N/A

          \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

       4. unpow2 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       5. lower-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

       6. lower-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   12. Applied rewrites 14.0%

       \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   13. Taylor expanded in eh around 0

       \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}}\right| \]
   14. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{1}{{ew}^{2}}}}}\right| \]

       2. pow-flip N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{{ew}^{\left(\mathsf{neg}\left(2\right)\right)}}}\right| \]

       3. metadata-eval N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{{ew}^{-2}}}\right| \]

       4. sqrt-pow1 N/A

          \[\leadsto \left|\frac{eh}{ew \cdot {ew}^{\left(\frac{-2}{\color{blue}{2}}\right)}}\right| \]

       5. metadata-eval N/A

          \[\leadsto \left|\frac{eh}{ew \cdot {ew}^{-1}}\right| \]

       6. inv-pow N/A

          \[\leadsto \left|\frac{eh}{ew \cdot \frac{1}{ew}}\right| \]

       7. lower-*.f64 N/A

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

       8. lower-/.f64 52.9

          \[\leadsto \left|\frac{eh}{ew \cdot \frac{1}{ew}}\right| \]

   15. Applied rewrites 52.9%

       \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \frac{1}{ew}}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 34.1% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \left|\frac{eh}{ew \cdot \frac{1}{ew}}\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (/ eh (* ew (/ 1.0 ew)))))

C

double code(double eh, double ew, double t) {
	return fabs((eh / (ew * (1.0 / 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((eh / (ew * (1.0d0 / ew))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((eh / (ew * (1.0 / ew))));
}

Python

def code(eh, ew, t):
	return math.fabs((eh / (ew * (1.0 / ew))))

Julia

function code(eh, ew, t)
	return abs(Float64(eh / Float64(ew * Float64(1.0 / ew))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((eh / (ew * (1.0 / ew))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(eh / N[(ew * N[(1.0 / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 89.7

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

7. Applied rewrites 89.7%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lower-fma.f64 N/A

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

9. Applied rewrites 57.5%

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

10. Taylor expanded in t around 0

    \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
11. Step-by-step derivation

    1. cos-atan-rev N/A

       \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

    2. sin-atan-rev N/A

       \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

    3. lower-/.f64 N/A

       \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

    4. unpow2 N/A

       \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

    5. lower-*.f64 N/A

       \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

    6. lower-*.f64 N/A

       \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

    7. lower-sqrt.f64 N/A

       \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

12. Applied rewrites 11.2%

    \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

13. Taylor expanded in eh around 0

    \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}}\right| \]
14. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \left|\frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{1}{{ew}^{2}}}}}\right| \]

    2. pow-flip N/A

       \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{{ew}^{\left(\mathsf{neg}\left(2\right)\right)}}}\right| \]

    3. metadata-eval N/A

       \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{{ew}^{-2}}}\right| \]

    4. sqrt-pow1 N/A

       \[\leadsto \left|\frac{eh}{ew \cdot {ew}^{\left(\frac{-2}{\color{blue}{2}}\right)}}\right| \]

    5. metadata-eval N/A

       \[\leadsto \left|\frac{eh}{ew \cdot {ew}^{-1}}\right| \]

    6. inv-pow N/A

       \[\leadsto \left|\frac{eh}{ew \cdot \frac{1}{ew}}\right| \]

    7. lower-*.f64 N/A

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

    8. lower-/.f64 42.1

       \[\leadsto \left|\frac{eh}{ew \cdot \frac{1}{ew}}\right| \]

15. Applied rewrites 42.1%

    \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \frac{1}{ew}}}\right| \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))