Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.9s

Alternatives: 6

Speedup: 1.2×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

6. Applied rewrites 98.4%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

8. Applied rewrites 98.4%

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

Alternative 3: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Taylor expanded in eh around 0

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

6. Applied rewrites 98.4%

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

Alternative 4: 98.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Taylor expanded in eh around 0

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

6. Applied rewrites 98.4%

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

7. Taylor expanded in t around 0

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

   1. lower-/.f64 98.1

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

9. Applied rewrites 98.1%

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

Alternative 5: 62.1% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

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 62.1

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

6. Applied rewrites 62.1%

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

Alternative 6: 20.7% accurate, 32.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{ew} \cdot \sqrt{ew} \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (* (sqrt ew) (sqrt ew)))

C

double code(double eh, double ew, double t) {
	return sqrt(ew) * sqrt(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 = sqrt(ew) * sqrt(ew)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 90.0

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

4. Applied rewrites 90.0%

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

5. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 90.0

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

7. Applied rewrites 90.0%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

9. Applied rewrites 43.3%

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

10. Taylor expanded in t around 0

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

    1. lower-sqrt.f64 19.0

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

12. Applied rewrites 19.0%

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

13. Taylor expanded in t around 0

    \[\leadsto \sqrt{ew} \cdot \color{blue}{\sqrt{ew}} \]
14. Step-by-step derivation

    1. lower-sqrt.f64 20.7

       \[\leadsto \sqrt{ew} \cdot \sqrt{ew} \]

15. Applied rewrites 20.7%

    \[\leadsto \sqrt{ew} \cdot \color{blue}{\sqrt{ew}} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2025151 
(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)))))))