Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 14.5s

Alternatives: 9

Speedup: 1.2×

Specification

Math

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

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

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

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

Alternative 2: 99.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

def code(eh, ew, t):
	return math.fabs((((math.cos(t) * ew) / math.sqrt((math.pow(((math.tan(t) * eh) / ew), 2.0) - -1.0))) - (math.tanh(math.asinh(((t / ew) * -eh))) * (math.sin(t) * eh))))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((cos(t) * ew) / sqrt(((((tan(t) * eh) / ew) ^ 2.0) - -1.0))) - (tanh(asinh(((t / ew) * -eh))) * (sin(t) * eh))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in t around 0

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

   1. lower-/.f64 99.0%

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

6. Applied rewrites 99.0%

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

Alternative 3: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in eh around 0

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

   1. lower-cos.f64 98.5%

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

8. Applied rewrites 98.5%

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

Alternative 4: 73.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|ew\right| \leq 370000000000:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{eh}{\left|ew\right|} \cdot \tan t\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{\left|ew\right|} \cdot \left(-eh\right)\right), \frac{1}{\sqrt{1}}, \frac{\cos t \cdot \left|ew\right|}{\sqrt{1}}\right)\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
  :precision binary64
  (if (<= (fabs ew) 370000000000.0)
  (fabs
   (* (tanh (asinh (* (/ eh (fabs ew)) (tan t)))) (* (sin t) eh)))
  (fabs
   (fma
    (* (* (sin t) (- eh)) (* (/ (tan t) (fabs ew)) (- eh)))
    (/ 1.0 (sqrt 1.0))
    (/ (* (cos t) (fabs ew)) (sqrt 1.0))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (fabs(ew) <= 370000000000.0) {
		tmp = fabs((tanh(asinh(((eh / fabs(ew)) * tan(t)))) * (sin(t) * eh)));
	} else {
		tmp = fabs(fma(((sin(t) * -eh) * ((tan(t) / fabs(ew)) * -eh)), (1.0 / sqrt(1.0)), ((cos(t) * fabs(ew)) / sqrt(1.0))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (abs(ew) <= 370000000000.0)
		tmp = abs(Float64(tanh(asinh(Float64(Float64(eh / abs(ew)) * tan(t)))) * Float64(sin(t) * eh)));
	else
		tmp = abs(fma(Float64(Float64(sin(t) * Float64(-eh)) * Float64(Float64(tan(t) / abs(ew)) * Float64(-eh))), Float64(1.0 / sqrt(1.0)), Float64(Float64(cos(t) * abs(ew)) / sqrt(1.0))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[N[Abs[ew], $MachinePrecision], 370000000000.0], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(eh / N[Abs[ew], $MachinePrecision]), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Sin[t], $MachinePrecision] * (-eh)), $MachinePrecision] * N[(N[(N[Tan[t], $MachinePrecision] / N[Abs[ew], $MachinePrecision]), $MachinePrecision] * (-eh)), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * N[Abs[ew], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 3.7e11

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 40.4%

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

   6. Applied rewrites 40.4%

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

   if 3.7e11 < ew

   1. Initial program 99.8%

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

   3. Applied rewrites 78.5%

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

   4. Taylor expanded in eh around 0

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

   6. Applied rewrites 63.4%

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

   7. Taylor expanded in eh around 0

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

   9. Applied rewrites 63.2%

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

Alternative 5: 72.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|ew\right| \leq 370000000000:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{eh}{\left|ew\right|} \cdot \tan t\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left|ew\right| \cdot \cos t\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
  :precision binary64
  (if (<= (fabs ew) 370000000000.0)
  (fabs
   (* (tanh (asinh (* (/ eh (fabs ew)) (tan t)))) (* (sin t) eh)))
  (fabs (* (fabs ew) (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (fabs(ew) <= 370000000000.0) {
		tmp = fabs((tanh(asinh(((eh / fabs(ew)) * tan(t)))) * (sin(t) * eh)));
	} else {
		tmp = fabs((fabs(ew) * cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if math.fabs(ew) <= 370000000000.0:
		tmp = math.fabs((math.tanh(math.asinh(((eh / math.fabs(ew)) * math.tan(t)))) * (math.sin(t) * eh)))
	else:
		tmp = math.fabs((math.fabs(ew) * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (abs(ew) <= 370000000000.0)
		tmp = abs(Float64(tanh(asinh(Float64(Float64(eh / abs(ew)) * tan(t)))) * Float64(sin(t) * eh)));
	else
		tmp = abs(Float64(abs(ew) * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (abs(ew) <= 370000000000.0)
		tmp = abs((tanh(asinh(((eh / abs(ew)) * tan(t)))) * (sin(t) * eh)));
	else
		tmp = abs((abs(ew) * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[N[Abs[ew], $MachinePrecision], 370000000000.0], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(eh / N[Abs[ew], $MachinePrecision]), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Abs[ew], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 3.7e11

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 40.4%

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

   6. Applied rewrites 40.4%

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

   if 3.7e11 < ew

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 62.8%

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

   6. Applied rewrites 62.8%

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

Alternative 6: 62.8% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((ew * cos(t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 62.8%

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

6. Applied rewrites 62.8%

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

Alternative 7: 41.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|t\right| \leq 8 \cdot 10^{+33}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left|t\right| \cdot \left|t\right|, -0.5 \cdot ew, ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{ew}^{2}}\\ \end{array} \]

FPCore

(FPCore (eh ew t)
  :precision binary64
  (if (<= (fabs t) 8e+33)
  (fabs (fma (* (fabs t) (fabs t)) (* -0.5 ew) ew))
  (sqrt (pow ew 2.0))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (fabs(t) <= 8e+33) {
		tmp = fabs(fma((fabs(t) * fabs(t)), (-0.5 * ew), ew));
	} else {
		tmp = sqrt(pow(ew, 2.0));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (abs(t) <= 8e+33)
		tmp = abs(fma(Float64(abs(t) * abs(t)), Float64(-0.5 * ew), ew));
	else
		tmp = sqrt((ew ^ 2.0));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[N[Abs[t], $MachinePrecision], 8e+33], N[Abs[N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(-0.5 * ew), $MachinePrecision] + ew), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Power[ew, 2.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.9999999999999996e33

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 62.8%

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

   6. Applied rewrites 62.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 39.5%

         \[\leadsto \left|ew + -0.5 \cdot \left(ew \cdot {t}^{2}\right)\right| \]

   9. Applied rewrites 39.5%

      \[\leadsto \left|ew + \color{blue}{-0.5 \cdot \left(ew \cdot {t}^{2}\right)}\right| \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 39.5%

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

   11. Applied rewrites 39.5%

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

   if 7.9999999999999996e33 < t

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 62.8%

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

   6. Applied rewrites 62.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\left(ew \cdot \cos t\right) \cdot \left(ew \cdot \cos t\right)\right)}^{\frac{1}{2}}} \]

      4. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\left(ew \cdot \cos t\right) \cdot \left(ew \cdot \cos t\right)\right) \cdot \frac{1}{2}}} \]

   8. Applied rewrites 32.8%

      \[\leadsto \color{blue}{e^{\log \left({\left(ew \cdot \cos t\right)}^{2}\right) \cdot 0.5}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{{ew}^{2}}} \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{{ew}^{2}} \]

       2. lower-pow.f64 25.6%

          \[\leadsto \sqrt{{ew}^{2}} \]

   11. Applied rewrites 25.6%

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

Alternative 8: 39.5% accurate, 19.9× speedup

Math

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

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 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 62.8%

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

6. Applied rewrites 62.8%

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

7. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 39.5%

      \[\leadsto \left|ew + -0.5 \cdot \left(ew \cdot {t}^{2}\right)\right| \]

9. Applied rewrites 39.5%

   \[\leadsto \left|ew + \color{blue}{-0.5 \cdot \left(ew \cdot {t}^{2}\right)}\right| \]
10. Step-by-step derivation

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. associate-*r* N/A

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

    6. lift-pow.f64 N/A

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

    7. unpow2 N/A

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

    8. associate-*r* N/A

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

    9. lower-fma.f64 N/A

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

    10. lower-*.f64 N/A

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

    11. lower-*.f64 39.5%

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

11. Applied rewrites 39.5%

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

Alternative 9: 39.5% accurate, 19.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 62.8%

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

6. Applied rewrites 62.8%

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

7. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 39.5%

      \[\leadsto \left|ew + -0.5 \cdot \left(ew \cdot {t}^{2}\right)\right| \]

9. Applied rewrites 39.5%

   \[\leadsto \left|ew + \color{blue}{-0.5 \cdot \left(ew \cdot {t}^{2}\right)}\right| \]
10. Step-by-step derivation

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f64 N/A

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

    8. lift-pow.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 N/A

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

    11. lower-*.f64 39.5%

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

11. Applied rewrites 39.5%

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

Reproduce

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