Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 13.5s

Alternatives: 8

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((((eh * sin(t)) * sin(atan(((eh * tan(t)) / -ew)))) - ((cos(atan(((tan(t) / ew) * eh))) * cos(t)) * 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 * sin(t)) * sin(atan(((eh * tan(t)) / -ew)))) - ((cos(atan(((tan(t) / ew) * eh))) * cos(t)) * ew)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 78.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ t_2 := \left|\left(eh \cdot \sin t\right) \cdot \sin t\_1 - \left(ew \cdot \cos t\right) \cdot \cos t\_1\right|\\ t_3 := \cos t \cdot ew\\ \mathbf{if}\;t\_2 \leq 10^{-105}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, \frac{\tan t}{ew} \cdot \left(eh \cdot eh\right), t\_3\right)}{1}\right|\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+129}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, 0\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_3\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew))))
        (t_2
         (fabs (- (* (* eh (sin t)) (sin t_1)) (* (* ew (cos t)) (cos t_1)))))
        (t_3 (* (cos t) ew)))
   (if (<= t_2 1e-105)
     (fabs (/ (fma (sin t) (* (/ (tan t) ew) (* eh eh)) t_3) 1.0))
     (if (<= t_2 5e+129)
       (fabs
        (fma
         (/ ew 1.0)
         (cos t)
         (* (* (- eh) (sin t)) (tanh (* t (fma -1.0 (/ eh ew) 0.0))))))
       (fabs t_3)))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double t_2 = fabs((((eh * sin(t)) * sin(t_1)) - ((ew * cos(t)) * cos(t_1))));
	double t_3 = cos(t) * ew;
	double tmp;
	if (t_2 <= 1e-105) {
		tmp = fabs((fma(sin(t), ((tan(t) / ew) * (eh * eh)), t_3) / 1.0));
	} else if (t_2 <= 5e+129) {
		tmp = fabs(fma((ew / 1.0), cos(t), ((-eh * sin(t)) * tanh((t * fma(-1.0, (eh / ew), 0.0))))));
	} else {
		tmp = fabs(t_3);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	t_2 = abs(Float64(Float64(Float64(eh * sin(t)) * sin(t_1)) - Float64(Float64(ew * cos(t)) * cos(t_1))))
	t_3 = Float64(cos(t) * ew)
	tmp = 0.0
	if (t_2 <= 1e-105)
		tmp = abs(Float64(fma(sin(t), Float64(Float64(tan(t) / ew) * Float64(eh * eh)), t_3) / 1.0));
	elseif (t_2 <= 5e+129)
		tmp = abs(fma(Float64(ew / 1.0), cos(t), Float64(Float64(Float64(-eh) * sin(t)) * tanh(Float64(t * fma(-1.0, Float64(eh / ew), 0.0))))));
	else
		tmp = abs(t_3);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-105], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh * eh), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+129], N[Abs[N[(N[(ew / 1.0), $MachinePrecision] * N[Cos[t], $MachinePrecision] + N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Tanh[N[(t * N[(-1.0 * N[(eh / ew), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[t$95$3], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (fabs.f64 (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))) < 9.99999999999999965e-106

   1. Initial program 99.8%

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

   4. Applied rewrites 87.0%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 74.3%

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

   if 9.99999999999999965e-106 < (fabs.f64 (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))) < 5.0000000000000003e129

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. sqr-neg-rev N/A

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

      14. cosh-asinh-rev N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 98.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 98.5%

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

   10. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. distribute-lft1-in N/A

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

       8. metadata-eval N/A

          \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(\frac{1}{2} \cdot t\right) \cdot \left(\color{blue}{0} \cdot \frac{{eh}^{2}}{{ew}^{2}}\right)\right)\right)\right)\right| \]

       9. lower-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(\frac{1}{2} \cdot t\right) \cdot \color{blue}{\left(0 \cdot \frac{{eh}^{2}}{{ew}^{2}}\right)}\right)\right)\right)\right| \]

       10. lower-/.f64 N/A

           \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(\frac{1}{2} \cdot t\right) \cdot \left(0 \cdot \color{blue}{\frac{{eh}^{2}}{{ew}^{2}}}\right)\right)\right)\right)\right| \]

       11. unpow2 N/A

           \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(\frac{1}{2} \cdot t\right) \cdot \left(0 \cdot \frac{\color{blue}{eh \cdot eh}}{{ew}^{2}}\right)\right)\right)\right)\right| \]

       12. lower-*.f64 N/A

           \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(\frac{1}{2} \cdot t\right) \cdot \left(0 \cdot \frac{\color{blue}{eh \cdot eh}}{{ew}^{2}}\right)\right)\right)\right)\right| \]

       13. unpow2 N/A

           \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(\frac{1}{2} \cdot t\right) \cdot \left(0 \cdot \frac{eh \cdot eh}{\color{blue}{ew \cdot ew}}\right)\right)\right)\right)\right| \]

       14. lower-*.f64 67.3

           \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(0.5 \cdot t\right) \cdot \left(0 \cdot \frac{eh \cdot eh}{\color{blue}{ew \cdot ew}}\right)\right)\right)\right)\right| \]

   12. Applied rewrites 67.3%

       \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \color{blue}{\left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(0.5 \cdot t\right) \cdot \left(0 \cdot \frac{eh \cdot eh}{ew \cdot ew}\right)\right)\right)}\right)\right| \]

   if 5.0000000000000003e129 < (fabs.f64 (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))))

   1. Initial program 99.7%

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 59.2

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

   7. Applied rewrites 59.2%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\right| \leq 10^{-105}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, \frac{\tan t}{ew} \cdot \left(eh \cdot eh\right), \cos t \cdot ew\right)}{1}\right|\\ \mathbf{elif}\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\right| \leq 5 \cdot 10^{+129}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, 0\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

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 * sin(t)) * sin(atan(((eh * tan(t)) / -ew)))) - ((cos(t) * ew) / 1.0d0)))
end function

Java

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

Python

def code(eh, ew, t):
	return math.fabs((((eh * math.sin(t)) * math.sin(math.atan(((eh * math.tan(t)) / -ew)))) - ((math.cos(t) * ew) / 1.0)))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((eh * sin(t)) * sin(atan(((eh * tan(t)) / -ew)))) - ((cos(t) * ew) / 1.0)));
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. *-lft-identity N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. sqr-neg-rev N/A

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

   14. cosh-asinh-rev N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eh around 0

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

7. Applied rewrites 98.6%

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

8. Final simplification 98.6%

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

Alternative 4: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-5} \lor \neg \left(t \leq 4 \cdot 10^{-7}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, 0\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-eh, \sin \tan^{-1} \left(\frac{\frac{\sin t \cdot eh}{-ew}}{\cos t}\right) \cdot t, ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -3.9e-5) (not (<= t 4e-7)))
   (fabs
    (fma
     (/ ew 1.0)
     (cos t)
     (* (* (- eh) (sin t)) (tanh (* t (fma -1.0 (/ eh ew) 0.0))))))
   (fabs
    (fma (- eh) (* (sin (atan (/ (/ (* (sin t) eh) (- ew)) (cos t)))) t) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -3.9e-5) || !(t <= 4e-7)) {
		tmp = fabs(fma((ew / 1.0), cos(t), ((-eh * sin(t)) * tanh((t * fma(-1.0, (eh / ew), 0.0))))));
	} else {
		tmp = fabs(fma(-eh, (sin(atan((((sin(t) * eh) / -ew) / cos(t)))) * t), ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -3.9e-5) || !(t <= 4e-7))
		tmp = abs(fma(Float64(ew / 1.0), cos(t), Float64(Float64(Float64(-eh) * sin(t)) * tanh(Float64(t * fma(-1.0, Float64(eh / ew), 0.0))))));
	else
		tmp = abs(fma(Float64(-eh), Float64(sin(atan(Float64(Float64(Float64(sin(t) * eh) / Float64(-ew)) / cos(t)))) * t), ew));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -3.9e-5], N[Not[LessEqual[t, 4e-7]], $MachinePrecision]], N[Abs[N[(N[(ew / 1.0), $MachinePrecision] * N[Cos[t], $MachinePrecision] + N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Tanh[N[(t * N[(-1.0 * N[(eh / ew), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[((-eh) * N[(N[Sin[N[ArcTan[N[(N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] / (-ew)), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] + ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.8999999999999999e-5 or 3.9999999999999998e-7 < t

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. sqr-neg-rev N/A

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

      14. cosh-asinh-rev N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 97.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 97.6%

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

   10. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. distribute-lft1-in N/A

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

       8. metadata-eval N/A

          \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(\frac{1}{2} \cdot t\right) \cdot \left(\color{blue}{0} \cdot \frac{{eh}^{2}}{{ew}^{2}}\right)\right)\right)\right)\right| \]

       9. lower-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(\frac{1}{2} \cdot t\right) \cdot \color{blue}{\left(0 \cdot \frac{{eh}^{2}}{{ew}^{2}}\right)}\right)\right)\right)\right| \]

       10. lower-/.f64 N/A

           \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(\frac{1}{2} \cdot t\right) \cdot \left(0 \cdot \color{blue}{\frac{{eh}^{2}}{{ew}^{2}}}\right)\right)\right)\right)\right| \]

       11. unpow2 N/A

           \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(\frac{1}{2} \cdot t\right) \cdot \left(0 \cdot \frac{\color{blue}{eh \cdot eh}}{{ew}^{2}}\right)\right)\right)\right)\right| \]

       12. lower-*.f64 N/A

           \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(\frac{1}{2} \cdot t\right) \cdot \left(0 \cdot \frac{\color{blue}{eh \cdot eh}}{{ew}^{2}}\right)\right)\right)\right)\right| \]

       13. unpow2 N/A

           \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(\frac{1}{2} \cdot t\right) \cdot \left(0 \cdot \frac{eh \cdot eh}{\color{blue}{ew \cdot ew}}\right)\right)\right)\right)\right| \]

       14. lower-*.f64 53.5

           \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(0.5 \cdot t\right) \cdot \left(0 \cdot \frac{eh \cdot eh}{\color{blue}{ew \cdot ew}}\right)\right)\right)\right)\right| \]

   12. Applied rewrites 53.5%

       \[\leadsto \left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \color{blue}{\left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, \left(0.5 \cdot t\right) \cdot \left(0 \cdot \frac{eh \cdot eh}{ew \cdot ew}\right)\right)\right)}\right)\right| \]

   if -3.8999999999999999e-5 < t < 3.9999999999999998e-7

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. sqr-neg-rev N/A

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

      14. cosh-asinh-rev N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-5} \lor \neg \left(t \leq 4 \cdot 10^{-7}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{ew}{1}, \cos t, \left(\left(-eh\right) \cdot \sin t\right) \cdot \tanh \left(t \cdot \mathsf{fma}\left(-1, \frac{eh}{ew}, 0\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-eh, \sin \tan^{-1} \left(\frac{\frac{\sin t \cdot eh}{-ew}}{\cos t}\right) \cdot t, ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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 * sin(t)) * sin(atan(((-eh * t) / ew)))) - ((cos(t) * ew) / 1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. *-lft-identity N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. sqr-neg-rev N/A

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

   14. cosh-asinh-rev N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eh around 0

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 98.2

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

10. Applied rewrites 98.2%

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

11. Final simplification 98.2%

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

Alternative 6: 60.6% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 67.9%

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

5. Taylor expanded in eh around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 59.7

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

7. Applied rewrites 59.7%

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

Alternative 7: 38.0% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(\left(-0.5 \cdot ew\right) \cdot t, t, ew\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (fma (* (* -0.5 ew) t) t ew)))

C

double code(double eh, double ew, double t) {
	return fabs(fma(((-0.5 * ew) * t), t, ew));
}

Julia

function code(eh, ew, t)
	return abs(fma(Float64(Float64(-0.5 * ew) * t), t, ew))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(-0.5 * ew), $MachinePrecision] * t), $MachinePrecision] * t + ew), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 67.9%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 30.9

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

7. Applied rewrites 30.9%

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

8. Taylor expanded in eh around 0

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

10. Applied rewrites 35.5%

    \[\leadsto \left|\mathsf{fma}\left(-0.5 \cdot ew, \color{blue}{t} \cdot t, ew\right)\right| \]
11. Step-by-step derivation

12. Applied rewrites 35.5%

    \[\leadsto \left|\mathsf{fma}\left(\left(-0.5 \cdot ew\right) \cdot t, \color{blue}{t}, ew\right)\right| \]
13. Add Preprocessing

Alternative 8: 37.9% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(-0.5 \cdot ew, t \cdot t, ew\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (fma (* -0.5 ew) (* t t) ew)))

C

double code(double eh, double ew, double t) {
	return fabs(fma((-0.5 * ew), (t * t), ew));
}

Julia

function code(eh, ew, t)
	return abs(fma(Float64(-0.5 * ew), Float64(t * t), ew))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(-0.5 * ew), $MachinePrecision] * N[(t * t), $MachinePrecision] + ew), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 67.9%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 30.9

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

7. Applied rewrites 30.9%

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

8. Taylor expanded in eh around 0

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

10. Applied rewrites 35.5%

    \[\leadsto \left|\mathsf{fma}\left(-0.5 \cdot ew, \color{blue}{t} \cdot t, ew\right)\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024351 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))