Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 11.7s

Alternatives: 10

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

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

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

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

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

3. Applied rewrites 99.8%

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

Alternative 2: 98.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-atan.f64 N/A

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

   6. cos-atan N/A

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

   7. mult-flip-rev N/A

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

   8. add-to-fraction N/A

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

3. Applied rewrites 65.1%

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

4. Taylor expanded in eh around 0

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

6. Applied rewrites 41.9%

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

7. Taylor expanded in eh around 0

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

9. Applied rewrites 98.6%

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

Alternative 3: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

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

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

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

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

6. Applied rewrites 98.6%

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

Alternative 4: 89.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_1 := \sinh^{-1} \left(\frac{eh}{t \cdot ew}\right)\\ \mathbf{if}\;eh \leq -8.2 \cdot 10^{+96}:\\ \;\;\;\;\left(-\left|\cos t\right|\right) \cdot eh\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\tanh t\_1 \cdot \cos t, eh, \frac{\sin t \cdot ew}{\cosh t\_1}\right)\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (asinh (/ eh (* t ew)))))
   (if (<= eh -8.2e+96)
     (* (- (fabs (cos t))) eh)
     (fabs (fma (* (tanh t_1) (cos t)) eh (/ (* (sin t) ew) (cosh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = asinh((eh / (t * ew)));
	double tmp;
	if (eh <= -8.2e+96) {
		tmp = -fabs(cos(t)) * eh;
	} else {
		tmp = fabs(fma((tanh(t_1) * cos(t)), eh, ((sin(t) * ew) / cosh(t_1))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = asinh(Float64(eh / Float64(t * ew)))
	tmp = 0.0
	if (eh <= -8.2e+96)
		tmp = Float64(Float64(-abs(cos(t))) * eh);
	else
		tmp = abs(fma(Float64(tanh(t_1) * cos(t)), eh, Float64(Float64(sin(t) * ew) / cosh(t_1))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcSinh[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -8.2e+96], N[((-N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]) * eh), $MachinePrecision], N[Abs[N[(N[(N[Tanh[t$95$1], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[Cosh[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.19999999999999996e96

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      5. lower-cos.f64 32.5

         \[\leadsto -1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

   8. Applied rewrites 32.5%

      \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{{\cos t}^{2}} \cdot eh\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{{\cos t}^{2}}\right)\right) \cdot \color{blue}{eh} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{{\cos t}^{2}}\right)\right) \cdot \color{blue}{eh} \]

      7. lower-neg.f64 32.5

         \[\leadsto \left(-\sqrt{{\cos t}^{2}}\right) \cdot eh \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{{\cos t}^{2}}\right) \cdot eh \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(-\sqrt{{\cos t}^{2}}\right) \cdot eh \]

      10. unpow2 N/A

          \[\leadsto \left(-\sqrt{\cos t \cdot \cos t}\right) \cdot eh \]

      11. rem-sqrt-square N/A

          \[\leadsto \left(-\left|\cos t\right|\right) \cdot eh \]

      12. lower-fabs.f64 32.5

          \[\leadsto \left(-\left|\cos t\right|\right) \cdot eh \]

   10. Applied rewrites 32.5%

       \[\leadsto \left(-\left|\cos t\right|\right) \cdot \color{blue}{eh} \]

   if -8.19999999999999996e96 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 89.4%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 89.5%

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

Alternative 5: 88.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;eh \leq -6.5 \cdot 10^{+82}:\\ \;\;\;\;\left(-\left|\cos t\right|\right) \cdot eh\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{\frac{1}{\mathsf{fma}\left(\left(\cos t \cdot eh\right) \cdot 1, \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right), \sin t \cdot ew\right)}}\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -6.5e+82)
   (* (- (fabs (cos t))) eh)
   (fabs
    (/
     1.0
     (/
      1.0
      (fma
       (* (* (cos t) eh) 1.0)
       (tanh (asinh (/ eh (* ew t))))
       (* (sin t) ew)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -6.5e+82) {
		tmp = -fabs(cos(t)) * eh;
	} else {
		tmp = fabs((1.0 / (1.0 / fma(((cos(t) * eh) * 1.0), tanh(asinh((eh / (ew * t)))), (sin(t) * ew)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -6.5e+82)
		tmp = Float64(Float64(-abs(cos(t))) * eh);
	else
		tmp = abs(Float64(1.0 / Float64(1.0 / fma(Float64(Float64(cos(t) * eh) * 1.0), tanh(asinh(Float64(eh / Float64(ew * t)))), Float64(sin(t) * ew)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -6.5e+82], N[((-N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]) * eh), $MachinePrecision], N[Abs[N[(1.0 / N[(1.0 / N[(N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * 1.0), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.5000000000000003e82

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      5. lower-cos.f64 32.5

         \[\leadsto -1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

   8. Applied rewrites 32.5%

      \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{{\cos t}^{2}} \cdot eh\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{{\cos t}^{2}}\right)\right) \cdot \color{blue}{eh} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{{\cos t}^{2}}\right)\right) \cdot \color{blue}{eh} \]

      7. lower-neg.f64 32.5

         \[\leadsto \left(-\sqrt{{\cos t}^{2}}\right) \cdot eh \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{{\cos t}^{2}}\right) \cdot eh \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(-\sqrt{{\cos t}^{2}}\right) \cdot eh \]

      10. unpow2 N/A

          \[\leadsto \left(-\sqrt{\cos t \cdot \cos t}\right) \cdot eh \]

      11. rem-sqrt-square N/A

          \[\leadsto \left(-\left|\cos t\right|\right) \cdot eh \]

      12. lower-fabs.f64 32.5

          \[\leadsto \left(-\left|\cos t\right|\right) \cdot eh \]

   10. Applied rewrites 32.5%

       \[\leadsto \left(-\left|\cos t\right|\right) \cdot \color{blue}{eh} \]

   if -6.5000000000000003e82 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. mult-flip-rev N/A

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

      8. add-to-fraction N/A

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

   3. Applied rewrites 65.1%

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

   4. Taylor expanded in eh around 0

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

   6. Applied rewrites 41.9%

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

   7. Taylor expanded in eh around 0

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

   9. Applied rewrites 98.6%

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

       1. lift-/.f64 N/A

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

       2. div-flip N/A

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

       3. lower-unsound-/.f64 N/A

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

       4. lower-unsound-/.f64 98.4

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

       5. lift-fma.f64 N/A

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

   11. Applied rewrites 98.4%

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

   12. Taylor expanded in t around 0

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

       1. lower-*.f64 88.8

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

   14. Applied rewrites 88.8%

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

Alternative 6: 74.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;eh \leq -1.5 \cdot 10^{-130}:\\ \;\;\;\;\left(-\left|\cos t\right|\right) \cdot eh\\ \mathbf{elif}\;eh \leq 1.5 \cdot 10^{-160}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;eh \cdot \sqrt{{\cos t}^{2}}\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -1.5e-130)
   (* (- (fabs (cos t))) eh)
   (if (<= eh 1.5e-160)
     (fabs (* (sin t) ew))
     (* eh (sqrt (pow (cos t) 2.0))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.5e-130) {
		tmp = -fabs(cos(t)) * eh;
	} else if (eh <= 1.5e-160) {
		tmp = fabs((sin(t) * ew));
	} else {
		tmp = eh * sqrt(pow(cos(t), 2.0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-1.5d-130)) then
        tmp = -abs(cos(t)) * eh
    else if (eh <= 1.5d-160) then
        tmp = abs((sin(t) * ew))
    else
        tmp = eh * sqrt((cos(t) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.5e-130) {
		tmp = -Math.abs(Math.cos(t)) * eh;
	} else if (eh <= 1.5e-160) {
		tmp = Math.abs((Math.sin(t) * ew));
	} else {
		tmp = eh * Math.sqrt(Math.pow(Math.cos(t), 2.0));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -1.5e-130:
		tmp = -math.fabs(math.cos(t)) * eh
	elif eh <= 1.5e-160:
		tmp = math.fabs((math.sin(t) * ew))
	else:
		tmp = eh * math.sqrt(math.pow(math.cos(t), 2.0))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -1.5e-130)
		tmp = Float64(Float64(-abs(cos(t))) * eh);
	elseif (eh <= 1.5e-160)
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = Float64(eh * sqrt((cos(t) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -1.5e-130)
		tmp = -abs(cos(t)) * eh;
	elseif (eh <= 1.5e-160)
		tmp = abs((sin(t) * ew));
	else
		tmp = eh * sqrt((cos(t) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -1.5e-130], N[((-N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]) * eh), $MachinePrecision], If[LessEqual[eh, 1.5e-160], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(eh * N[Sqrt[N[Power[N[Cos[t], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.49999999999999993e-130

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      5. lower-cos.f64 32.5

         \[\leadsto -1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

   8. Applied rewrites 32.5%

      \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{{\cos t}^{2}} \cdot eh\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{{\cos t}^{2}}\right)\right) \cdot \color{blue}{eh} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{{\cos t}^{2}}\right)\right) \cdot \color{blue}{eh} \]

      7. lower-neg.f64 32.5

         \[\leadsto \left(-\sqrt{{\cos t}^{2}}\right) \cdot eh \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{{\cos t}^{2}}\right) \cdot eh \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(-\sqrt{{\cos t}^{2}}\right) \cdot eh \]

      10. unpow2 N/A

          \[\leadsto \left(-\sqrt{\cos t \cdot \cos t}\right) \cdot eh \]

      11. rem-sqrt-square N/A

          \[\leadsto \left(-\left|\cos t\right|\right) \cdot eh \]

      12. lower-fabs.f64 32.5

          \[\leadsto \left(-\left|\cos t\right|\right) \cdot eh \]

   10. Applied rewrites 32.5%

       \[\leadsto \left(-\left|\cos t\right|\right) \cdot \color{blue}{eh} \]

   if -1.49999999999999993e-130 < eh < 1.49999999999999998e-160

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 41.8

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

   6. Applied rewrites 41.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 41.8

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

   8. Applied rewrites 41.8%

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

   if 1.49999999999999998e-160 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in eh around inf

      \[\leadsto \color{blue}{eh \cdot \sqrt{{\cos t}^{2}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto eh \cdot \color{blue}{\sqrt{{\cos t}^{2}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto eh \cdot \sqrt{{\cos t}^{2}} \]

      3. lower-pow.f64 N/A

         \[\leadsto eh \cdot \sqrt{{\cos t}^{2}} \]

      4. lower-cos.f64 31.3

         \[\leadsto eh \cdot \sqrt{{\cos t}^{2}} \]

   8. Applied rewrites 31.3%

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

Alternative 7: 64.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;eh \leq -1.5 \cdot 10^{-130}:\\ \;\;\;\;\left(-\left|\cos t\right|\right) \cdot eh\\ \mathbf{elif}\;eh \leq 1.5 \cdot 10^{-160}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -1.5e-130)
   (* (- (fabs (cos t))) eh)
   (if (<= eh 1.5e-160) (fabs (* (sin t) ew)) (fabs eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.5e-130) {
		tmp = -fabs(cos(t)) * eh;
	} else if (eh <= 1.5e-160) {
		tmp = fabs((sin(t) * ew));
	} else {
		tmp = fabs(eh);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-1.5d-130)) then
        tmp = -abs(cos(t)) * eh
    else if (eh <= 1.5d-160) then
        tmp = abs((sin(t) * ew))
    else
        tmp = abs(eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.5e-130) {
		tmp = -Math.abs(Math.cos(t)) * eh;
	} else if (eh <= 1.5e-160) {
		tmp = Math.abs((Math.sin(t) * ew));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -1.5e-130:
		tmp = -math.fabs(math.cos(t)) * eh
	elif eh <= 1.5e-160:
		tmp = math.fabs((math.sin(t) * ew))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -1.5e-130)
		tmp = Float64(Float64(-abs(cos(t))) * eh);
	elseif (eh <= 1.5e-160)
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = abs(eh);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -1.5e-130)
		tmp = -abs(cos(t)) * eh;
	elseif (eh <= 1.5e-160)
		tmp = abs((sin(t) * ew));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -1.5e-130], N[((-N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]) * eh), $MachinePrecision], If[LessEqual[eh, 1.5e-160], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[eh], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.49999999999999993e-130

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      5. lower-cos.f64 32.5

         \[\leadsto -1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

   8. Applied rewrites 32.5%

      \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sqrt{{\cos t}^{2}}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(eh \cdot \sqrt{{\cos t}^{2}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{{\cos t}^{2}} \cdot eh\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{{\cos t}^{2}}\right)\right) \cdot \color{blue}{eh} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{{\cos t}^{2}}\right)\right) \cdot \color{blue}{eh} \]

      7. lower-neg.f64 32.5

         \[\leadsto \left(-\sqrt{{\cos t}^{2}}\right) \cdot eh \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{{\cos t}^{2}}\right) \cdot eh \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(-\sqrt{{\cos t}^{2}}\right) \cdot eh \]

      10. unpow2 N/A

          \[\leadsto \left(-\sqrt{\cos t \cdot \cos t}\right) \cdot eh \]

      11. rem-sqrt-square N/A

          \[\leadsto \left(-\left|\cos t\right|\right) \cdot eh \]

      12. lower-fabs.f64 32.5

          \[\leadsto \left(-\left|\cos t\right|\right) \cdot eh \]

   10. Applied rewrites 32.5%

       \[\leadsto \left(-\left|\cos t\right|\right) \cdot \color{blue}{eh} \]

   if -1.49999999999999993e-130 < eh < 1.49999999999999998e-160

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 41.8

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

   6. Applied rewrites 41.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 41.8

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

   8. Applied rewrites 41.8%

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

   if 1.49999999999999998e-160 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 25.2

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

   8. Applied rewrites 25.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. rem-sqrt-square N/A

         \[\leadsto \left|eh\right| \]

      5. lower-fabs.f64 42.4

         \[\leadsto \left|eh\right| \]

   10. Applied rewrites 42.4%

       \[\leadsto \left|eh\right| \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;ew \leq -1.25 \cdot 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 2.32 \cdot 10^{+139}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= ew -1.25e-25) t_1 (if (<= ew 2.32e+139) (fabs eh) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (ew <= -1.25e-25) {
		tmp = t_1;
	} else if (ew <= 2.32e+139) {
		tmp = fabs(eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(t) * ew))
    if (ew <= (-1.25d-25)) then
        tmp = t_1
    else if (ew <= 2.32d+139) then
        tmp = abs(eh)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(t) * ew));
	double tmp;
	if (ew <= -1.25e-25) {
		tmp = t_1;
	} else if (ew <= 2.32e+139) {
		tmp = Math.abs(eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * ew))
	tmp = 0
	if ew <= -1.25e-25:
		tmp = t_1
	elif ew <= 2.32e+139:
		tmp = math.fabs(eh)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (ew <= -1.25e-25)
		tmp = t_1;
	elseif (ew <= 2.32e+139)
		tmp = abs(eh);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * ew));
	tmp = 0.0;
	if (ew <= -1.25e-25)
		tmp = t_1;
	elseif (ew <= 2.32e+139)
		tmp = abs(eh);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.25e-25], t$95$1, If[LessEqual[ew, 2.32e+139], N[Abs[eh], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.2499999999999999e-25 or 2.32000000000000008e139 < ew

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 41.8

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

   6. Applied rewrites 41.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 41.8

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

   8. Applied rewrites 41.8%

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

   if -1.2499999999999999e-25 < ew < 2.32000000000000008e139

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 25.2

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

   8. Applied rewrites 25.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. rem-sqrt-square N/A

         \[\leadsto \left|eh\right| \]

      5. lower-fabs.f64 42.4

         \[\leadsto \left|eh\right| \]

   10. Applied rewrites 42.4%

       \[\leadsto \left|eh\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 44.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;eh \leq -1.5 \cdot 10^{-130}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{elif}\;eh \leq 1.5 \cdot 10^{-160}:\\ \;\;\;\;\left|ew \cdot \left(t \cdot \left(1 + -0.16666666666666666 \cdot {t}^{2}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -1.5e-130)
   (fabs eh)
   (if (<= eh 1.5e-160)
     (fabs (* ew (* t (+ 1.0 (* -0.16666666666666666 (pow t 2.0))))))
     (fabs eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.5e-130) {
		tmp = fabs(eh);
	} else if (eh <= 1.5e-160) {
		tmp = fabs((ew * (t * (1.0 + (-0.16666666666666666 * pow(t, 2.0))))));
	} else {
		tmp = fabs(eh);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-1.5d-130)) then
        tmp = abs(eh)
    else if (eh <= 1.5d-160) then
        tmp = abs((ew * (t * (1.0d0 + ((-0.16666666666666666d0) * (t ** 2.0d0))))))
    else
        tmp = abs(eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.5e-130) {
		tmp = Math.abs(eh);
	} else if (eh <= 1.5e-160) {
		tmp = Math.abs((ew * (t * (1.0 + (-0.16666666666666666 * Math.pow(t, 2.0))))));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -1.5e-130:
		tmp = math.fabs(eh)
	elif eh <= 1.5e-160:
		tmp = math.fabs((ew * (t * (1.0 + (-0.16666666666666666 * math.pow(t, 2.0))))))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -1.5e-130)
		tmp = abs(eh);
	elseif (eh <= 1.5e-160)
		tmp = abs(Float64(ew * Float64(t * Float64(1.0 + Float64(-0.16666666666666666 * (t ^ 2.0))))));
	else
		tmp = abs(eh);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -1.5e-130)
		tmp = abs(eh);
	elseif (eh <= 1.5e-160)
		tmp = abs((ew * (t * (1.0 + (-0.16666666666666666 * (t ^ 2.0))))));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -1.5e-130], N[Abs[eh], $MachinePrecision], If[LessEqual[eh, 1.5e-160], N[Abs[N[(ew * N[(t * N[(1.0 + N[(-0.16666666666666666 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[eh], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.49999999999999993e-130 or 1.49999999999999998e-160 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 25.2

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

   8. Applied rewrites 25.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. rem-sqrt-square N/A

         \[\leadsto \left|eh\right| \]

      5. lower-fabs.f64 42.4

         \[\leadsto \left|eh\right| \]

   10. Applied rewrites 42.4%

       \[\leadsto \left|eh\right| \]

   if -1.49999999999999993e-130 < eh < 1.49999999999999998e-160

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 41.8

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

   6. Applied rewrites 41.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {t}^{2}}\right)\right)\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{t}^{2}}\right)\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \frac{-1}{6} \cdot {t}^{\color{blue}{2}}\right)\right)\right| \]

      4. lower-pow.f64 18.1

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + -0.16666666666666666 \cdot {t}^{2}\right)\right)\right| \]

   9. Applied rewrites 18.1%

      \[\leadsto \left|ew \cdot \left(t \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {t}^{2}\right)}\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 42.4% accurate, 112.2× speedup

Math

\[\left|eh\right| \]

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(eh)
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[eh], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

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

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

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

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 33.9%

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

6. Taylor expanded in t around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-pow.f64 25.2

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

8. Applied rewrites 25.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. rem-sqrt-square N/A

      \[\leadsto \left|eh\right| \]

   5. lower-fabs.f64 42.4

      \[\leadsto \left|eh\right| \]

10. Applied rewrites 42.4%

    \[\leadsto \left|eh\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025171 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))