Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 9.2s

Alternatives: 8

Speedup: N/A×

Specification

Math

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

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

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

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 98.3%

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

6. Applied rewrites 98.3%

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

Alternative 3: 89.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]}, N[Abs[N[(N[(N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $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. lower-*.f64 99.1%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 89.9%

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

7. Applied rewrites 89.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

9. Applied rewrites 89.9%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/r* N/A

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

    4. lower-/.f64 N/A

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

    5. lift-/.f64 89.9%

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

11. Applied rewrites 89.9%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/r* N/A

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

    4. lower-/.f64 N/A

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

    5. lift-/.f64 89.9%

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

13. Applied rewrites 89.9%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/r* N/A

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

    4. lower-/.f64 N/A

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

    5. lift-/.f64 89.9%

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

15. Applied rewrites 89.9%

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

Alternative 4: 89.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $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. lower-*.f64 99.1%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 89.9%

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

7. Applied rewrites 89.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

9. Applied rewrites 89.9%

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

Alternative 5: 65.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{\frac{eh}{t}}{ew}\\ t_2 := \frac{eh}{ew \cdot t}\\ t_3 := \left|\mathsf{fma}\left(\tanh \sinh^{-1} t\_2 \cdot \cos t, eh, \frac{t \cdot \left(ew + -0.16666666666666666 \cdot \left(ew \cdot {t}^{2}\right)\right)}{\sqrt{\mathsf{fma}\left(t\_2, t\_2, 1\right)}}\right)\right|\\ \mathbf{if}\;eh \leq -2.4 \cdot 10^{+42}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq 5.6 \cdot 10^{-33}:\\ \;\;\;\;\left|\mathsf{fma}\left(\tanh \sinh^{-1} t\_1 \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (eh ew t)
  :precision binary64
  (let* ((t_1 (/ (/ eh t) ew))
       (t_2 (/ eh (* ew t)))
       (t_3
        (fabs
         (fma
          (* (tanh (asinh t_2)) (cos t))
          eh
          (/
           (* t (+ ew (* -0.16666666666666666 (* ew (pow t 2.0)))))
           (sqrt (fma t_2 t_2 1.0)))))))
  (if (<= eh -2.4e+42)
    t_3
    (if (<= eh 5.6e-33)
      (fabs
       (fma
        (* (tanh (asinh t_1)) (+ 1.0 (* -0.5 (pow t 2.0))))
        eh
        (/ (* (sin t) ew) (sqrt (fma t_1 t_1 1.0)))))
      t_3))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / t) / ew;
	double t_2 = eh / (ew * t);
	double t_3 = fabs(fma((tanh(asinh(t_2)) * cos(t)), eh, ((t * (ew + (-0.16666666666666666 * (ew * pow(t, 2.0))))) / sqrt(fma(t_2, t_2, 1.0)))));
	double tmp;
	if (eh <= -2.4e+42) {
		tmp = t_3;
	} else if (eh <= 5.6e-33) {
		tmp = fabs(fma((tanh(asinh(t_1)) * (1.0 + (-0.5 * pow(t, 2.0)))), eh, ((sin(t) * ew) / sqrt(fma(t_1, t_1, 1.0)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / t) / ew)
	t_2 = Float64(eh / Float64(ew * t))
	t_3 = abs(fma(Float64(tanh(asinh(t_2)) * cos(t)), eh, Float64(Float64(t * Float64(ew + Float64(-0.16666666666666666 * Float64(ew * (t ^ 2.0))))) / sqrt(fma(t_2, t_2, 1.0)))))
	tmp = 0.0
	if (eh <= -2.4e+42)
		tmp = t_3;
	elseif (eh <= 5.6e-33)
		tmp = abs(fma(Float64(tanh(asinh(t_1)) * Float64(1.0 + Float64(-0.5 * (t ^ 2.0)))), eh, Float64(Float64(sin(t) * ew) / sqrt(fma(t_1, t_1, 1.0)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(N[(N[Tanh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh + N[(N[(t * N[(ew + N[(-0.16666666666666666 * N[(ew * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t$95$2 * t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.4e+42], t$95$3, If[LessEqual[eh, 5.6e-33], N[Abs[N[(N[(N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eh + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.3999999999999999e42 or 5.5999999999999998e-33 < 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. lower-*.f64 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 89.9%

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

   7. Applied rewrites 89.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   9. Applied rewrites 89.9%

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

   10. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \cos t, eh, \frac{t \cdot \color{blue}{\left(ew + \frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]

       2. lower-+.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \cos t, eh, \frac{t \cdot \left(ew + \color{blue}{\frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right)}\right)}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]

       3. lower-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \cos t, eh, \frac{t \cdot \left(ew + \frac{-1}{6} \cdot \color{blue}{\left(ew \cdot {t}^{2}\right)}\right)}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]

       4. lower-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \cos t, eh, \frac{t \cdot \left(ew + \frac{-1}{6} \cdot \left(ew \cdot \color{blue}{{t}^{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]

       5. lower-pow.f64 58.9%

          \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \cos t, eh, \frac{t \cdot \left(ew + -0.16666666666666666 \cdot \left(ew \cdot {t}^{\color{blue}{2}}\right)\right)}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]

   12. Applied rewrites 58.9%

       \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \cos t, eh, \frac{\color{blue}{t \cdot \left(ew + -0.16666666666666666 \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]

   if -2.3999999999999999e42 < eh < 5.5999999999999998e-33

   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. lower-*.f64 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 89.9%

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

   7. Applied rewrites 89.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   9. Applied rewrites 89.9%

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

   10. Taylor expanded in t around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 62.8%

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

   12. Applied rewrites 62.8%

       \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {t}^{2}\right)}, eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]
   13. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/r* N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 62.8%

          \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]

   14. Applied rewrites 62.8%

       \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{\color{blue}{ew}}\right) \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]
   15. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/r* N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 62.8%

          \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{\frac{eh}{t}}{ew}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]

   16. Applied rewrites 62.8%

       \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{\frac{eh}{t}}{\color{blue}{ew}}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]
   17. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/r* N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 62.8%

          \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{\frac{eh}{t}}{ew}, \frac{\frac{eh}{t}}{ew}, 1\right)}}\right)\right| \]

   18. Applied rewrites 62.8%

       \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{\frac{eh}{t}}{ew}, \frac{\frac{eh}{t}}{\color{blue}{ew}}, 1\right)}}\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 63.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{\frac{eh}{t}}{ew}\\ \left|\mathsf{fma}\left(\tanh \sinh^{-1} t\_1 \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
  :precision binary64
  (let* ((t_1 (/ (/ eh t) ew)))
  (fabs
   (fma
    (* (tanh (asinh t_1)) (+ 1.0 (* -0.5 (pow t 2.0))))
    eh
    (/ (* (sin t) ew) (sqrt (fma t_1 t_1 1.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / t) / ew;
	return fabs(fma((tanh(asinh(t_1)) * (1.0 + (-0.5 * pow(t, 2.0)))), eh, ((sin(t) * ew) / sqrt(fma(t_1, t_1, 1.0)))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / t) / ew)
	return abs(fma(Float64(tanh(asinh(t_1)) * Float64(1.0 + Float64(-0.5 * (t ^ 2.0)))), eh, Float64(Float64(sin(t) * ew) / sqrt(fma(t_1, t_1, 1.0)))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision]}, N[Abs[N[(N[(N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eh + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $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. lower-*.f64 99.1%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 89.9%

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

7. Applied rewrites 89.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

9. Applied rewrites 89.9%

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

10. Taylor expanded in t around 0

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

    1. lower-+.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-pow.f64 62.8%

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

12. Applied rewrites 62.8%

    \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {t}^{2}\right)}, eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]
13. Step-by-step derivation

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-/r* N/A

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

    5. lower-/.f64 N/A

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

    6. lower-/.f64 62.8%

       \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]

14. Applied rewrites 62.8%

    \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{\color{blue}{ew}}\right) \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]
15. Step-by-step derivation

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-/r* N/A

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

    5. lower-/.f64 N/A

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

    6. lower-/.f64 62.8%

       \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{\frac{eh}{t}}{ew}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]

16. Applied rewrites 62.8%

    \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{\frac{eh}{t}}{\color{blue}{ew}}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]
17. Step-by-step derivation

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-/r* N/A

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

    5. lower-/.f64 N/A

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

    6. lower-/.f64 62.8%

       \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{\frac{eh}{t}}{ew}, \frac{\frac{eh}{t}}{ew}, 1\right)}}\right)\right| \]

18. Applied rewrites 62.8%

    \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot \left(1 + -0.5 \cdot {t}^{2}\right), eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{\frac{eh}{t}}{ew}, \frac{\frac{eh}{t}}{\color{blue}{ew}}, 1\right)}}\right)\right| \]
19. Add Preprocessing

Alternative 7: 62.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ \left|\mathsf{fma}\left(eh \cdot \tanh \sinh^{-1} t\_1, \mathsf{fma}\left(t \cdot t, -0.5, 1\right), \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
  :precision binary64
  (let* ((t_1 (/ eh (* ew t))))
  (fabs
   (fma
    (* eh (tanh (asinh t_1)))
    (fma (* t t) -0.5 1.0)
    (/ (* (sin t) ew) (sqrt (fma t_1 t_1 1.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	return fabs(fma((eh * tanh(asinh(t_1))), fma((t * t), -0.5, 1.0), ((sin(t) * ew) / sqrt(fma(t_1, t_1, 1.0)))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * t))
	return abs(fma(Float64(eh * tanh(asinh(t_1))), fma(Float64(t * t), -0.5, 1.0), Float64(Float64(sin(t) * ew) / sqrt(fma(t_1, t_1, 1.0)))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(eh * N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $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. lower-*.f64 99.1%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 89.9%

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

7. Applied rewrites 89.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

9. Applied rewrites 89.9%

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

10. Taylor expanded in t around 0

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

    1. lower-+.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-pow.f64 62.8%

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

12. Applied rewrites 62.8%

    \[\leadsto \left|\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {t}^{2}\right)}, eh, \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)\right| \]
13. Step-by-step derivation

    1. lift-fma.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. associate-*r* N/A

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

    5. lower-fma.f64 N/A

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

14. Applied rewrites 63.0%

    \[\leadsto \left|\color{blue}{\mathsf{fma}\left(eh \cdot \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right), \mathsf{fma}\left(t \cdot t, -0.5, 1\right), \frac{\sin t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}\right)}\right| \]
15. Add Preprocessing

Alternative 8: 11.5% accurate, 3.9× speedup

Math

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

FPCore

(FPCore (eh ew t)
  :precision binary64
  (fabs (/ (pow eh 2.0) (* ew (sqrt (/ (pow eh 2.0) (pow ew 2.0)))))))

C

double code(double eh, double ew, double t) {
	return fabs((pow(eh, 2.0) / (ew * sqrt((pow(eh, 2.0) / pow(ew, 2.0))))));
}

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 ** 2.0d0) / (ew * sqrt(((eh ** 2.0d0) / (ew ** 2.0d0))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((Math.pow(eh, 2.0) / (ew * Math.sqrt((Math.pow(eh, 2.0) / Math.pow(ew, 2.0))))));
}

Python

def code(eh, ew, t):
	return math.fabs((math.pow(eh, 2.0) / (ew * math.sqrt((math.pow(eh, 2.0) / math.pow(ew, 2.0))))))

Julia

function code(eh, ew, t)
	return abs(Float64((eh ^ 2.0) / Float64(ew * sqrt(Float64((eh ^ 2.0) / (ew ^ 2.0))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((eh ^ 2.0) / (ew * sqrt(((eh ^ 2.0) / (ew ^ 2.0))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Power[eh, 2.0], $MachinePrecision] / N[(ew * N[Sqrt[N[(N[Power[eh, 2.0], $MachinePrecision] / N[Power[ew, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 59.5%

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

6. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 11.5%

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

8. Applied rewrites 11.5%

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

Reproduce

herbie shell --seed 2025212 
(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))))))))