Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 9.8s

Alternatives: 11

Speedup: N/A×

Specification

Math

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

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

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

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 3: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 98.4

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

6. Applied rewrites 98.4%

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

Alternative 4: 89.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \left(\log \left(2 \cdot \frac{eh}{ew}\right) + -1 \cdot \log t\right), ew \cdot \sin t\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (fma
   (* (cos t) eh)
   (tanh (+ (log (* 2.0 (/ eh ew))) (* -1.0 (log t))))
   (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	return fabs(fma((cos(t) * eh), tanh((log((2.0 * (eh / ew))) + (-1.0 * log(t)))), (ew * sin(t))));
}

Julia

function code(eh, ew, t)
	return abs(fma(Float64(cos(t) * eh), tanh(Float64(log(Float64(2.0 * Float64(eh / ew))) + Float64(-1.0 * log(t)))), Float64(ew * sin(t))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 98.4

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

6. Applied rewrites 98.4%

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

7. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-log.f64 27.6

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

9. Applied rewrites 27.6%

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

Alternative 5: 61.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 98.4

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

6. Applied rewrites 98.4%

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

7. Taylor expanded in t around 0

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

9. Applied rewrites 89.5%

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

Alternative 6: 53.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 1.28 \cdot 10^{-208}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right), \frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{eh}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot 1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 1.28e-208)
   (fabs
    (fma
     (* (cos t) eh)
     (tanh (asinh (/ eh (* ew t))))
     (/ (* (* ew ew) (* t t)) eh)))
   (fabs
    (fma
     eh
     (tanh
      (asinh (/ (fma -0.3333333333333333 (/ (* eh (* t t)) ew) (/ eh ew)) t)))
     (* (* (sin t) ew) 1.0)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 1.28e-208) {
		tmp = fabs(fma((cos(t) * eh), tanh(asinh((eh / (ew * t)))), (((ew * ew) * (t * t)) / eh)));
	} else {
		tmp = fabs(fma(eh, tanh(asinh((fma(-0.3333333333333333, ((eh * (t * t)) / ew), (eh / ew)) / t))), ((sin(t) * ew) * 1.0)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 1.28e-208)
		tmp = abs(fma(Float64(cos(t) * eh), tanh(asinh(Float64(eh / Float64(ew * t)))), Float64(Float64(Float64(ew * ew) * Float64(t * t)) / eh)));
	else
		tmp = abs(fma(eh, tanh(asinh(Float64(fma(-0.3333333333333333, Float64(Float64(eh * Float64(t * t)) / ew), Float64(eh / ew)) / t))), Float64(Float64(sin(t) * ew) * 1.0)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 1.28e-208], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(N[(N[(ew * ew), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Tanh[N[ArcSinh[N[(N[(-0.3333333333333333 * N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 1.2800000000000001e-208

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}, \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{\color{blue}{t}}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      3. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      5. pow2 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      7. lift-/.f64 88.8

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

   6. Applied rewrites 88.8%

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}, \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

   7. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}}^{2}}}\right)\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{\color{blue}{t}}\right)}^{2}}}\right)\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      3. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      5. pow2 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      7. lift-/.f64 76.2

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

   9. Applied rewrites 76.2%

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}}^{2}}}\right)\right| \]

   10. Taylor expanded in t around 0

       \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \color{blue}{\frac{{ew}^{2} \cdot {t}^{2}}{eh}}\right)\right| \]
   11. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \frac{\left(ew \cdot ew\right) \cdot {t}^{2}}{eh}\right)\right| \]

       2. pow2 N/A

          \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{eh}\right)\right| \]

       3. lift-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{eh}\right)\right| \]

       4. lift-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{eh}\right)\right| \]

       5. lift-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{eh}\right)\right| \]

       6. lift-/.f64 41.6

          \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{\color{blue}{eh}}\right)\right| \]

   12. Applied rewrites 41.6%

       \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \color{blue}{\frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{eh}}\right)\right| \]

   13. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lift-*.f64 46.7

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

   15. Applied rewrites 46.7%

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

   if 1.2800000000000001e-208 < ew

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}, \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{\color{blue}{t}}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      3. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      5. pow2 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      7. lift-/.f64 95.4

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

   6. Applied rewrites 95.4%

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}, \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

   7. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}}^{2}}}\right)\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{\color{blue}{t}}\right)}^{2}}}\right)\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      3. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      5. pow2 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      7. lift-/.f64 79.6

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

   9. Applied rewrites 79.6%

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}}^{2}}}\right)\right| \]

   10. Taylor expanded in t around 0

       \[\leadsto \left|\mathsf{fma}\left(\color{blue}{eh}, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 67.0%

       \[\leadsto \left|\mathsf{fma}\left(\color{blue}{eh}, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

   13. Taylor expanded in eh around 0

       \[\leadsto \left|\mathsf{fma}\left(eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \color{blue}{1}\right)\right| \]
   14. Step-by-step derivation

   15. Applied rewrites 82.2%

       \[\leadsto \left|\mathsf{fma}\left(eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \color{blue}{1}\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 30.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 980:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right), \frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{eh}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 980.0)
   (fabs
    (fma
     (* (cos t) eh)
     (tanh (asinh (/ eh (* ew t))))
     (/ (* (* ew ew) (* t t)) eh)))
   (fabs (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 980.0) {
		tmp = fabs(fma((cos(t) * eh), tanh(asinh((eh / (ew * t)))), (((ew * ew) * (t * t)) / eh)));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 980.0)
		tmp = abs(fma(Float64(cos(t) * eh), tanh(asinh(Float64(eh / Float64(ew * t)))), Float64(Float64(Float64(ew * ew) * Float64(t * t)) / eh)));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 980.0], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(N[(N[(ew * ew), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 980

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}, \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{\color{blue}{t}}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      3. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      5. pow2 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

      7. lift-/.f64 89.1

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

   6. Applied rewrites 89.1%

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}, \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}\right)\right| \]

   7. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}}^{2}}}\right)\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{\color{blue}{t}}\right)}^{2}}}\right)\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      3. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      5. pow2 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

      7. lift-/.f64 77.1

         \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}}\right)\right| \]

   9. Applied rewrites 77.1%

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}}^{2}}}\right)\right| \]

   10. Taylor expanded in t around 0

       \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \color{blue}{\frac{{ew}^{2} \cdot {t}^{2}}{eh}}\right)\right| \]
   11. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \frac{\left(ew \cdot ew\right) \cdot {t}^{2}}{eh}\right)\right| \]

       2. pow2 N/A

          \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{eh}\right)\right| \]

       3. lift-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{eh}\right)\right| \]

       4. lift-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{eh}\right)\right| \]

       5. lift-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{eh}\right)\right| \]

       6. lift-/.f64 44.9

          \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{\color{blue}{eh}}\right)\right| \]

   12. Applied rewrites 44.9%

       \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right), \color{blue}{\frac{\left(ew \cdot ew\right) \cdot \left(t \cdot t\right)}{eh}}\right)\right| \]

   13. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lift-*.f64 49.8

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

   15. Applied rewrites 49.8%

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

   if 980 < ew

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 64.5

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

   6. Applied rewrites 64.5%

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

Alternative 8: 27.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-17}:\\ \;\;\;\;\left|\tanh \left(\log \left(2 \cdot \frac{eh}{ew}\right) + -1 \cdot \log t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t 5.2e-17)
   (fabs (* (tanh (+ (log (* 2.0 (/ eh ew))) (* -1.0 (log t)))) eh))
   (fabs (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= 5.2e-17) {
		tmp = fabs((tanh((log((2.0 * (eh / ew))) + (-1.0 * log(t)))) * eh));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	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 (t <= 5.2d-17) then
        tmp = abs((tanh((log((2.0d0 * (eh / ew))) + ((-1.0d0) * log(t)))) * eh))
    else
        tmp = abs((ew * sin(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (t <= 5.2e-17) {
		tmp = Math.abs((Math.tanh((Math.log((2.0 * (eh / ew))) + (-1.0 * Math.log(t)))) * eh));
	} else {
		tmp = Math.abs((ew * Math.sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if t <= 5.2e-17:
		tmp = math.fabs((math.tanh((math.log((2.0 * (eh / ew))) + (-1.0 * math.log(t)))) * eh))
	else:
		tmp = math.fabs((ew * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= 5.2e-17)
		tmp = abs(Float64(tanh(Float64(log(Float64(2.0 * Float64(eh / ew))) + Float64(-1.0 * log(t)))) * eh));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (t <= 5.2e-17)
		tmp = abs((tanh((log((2.0 * (eh / ew))) + (-1.0 * log(t)))) * eh));
	else
		tmp = abs((ew * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, 5.2e-17], N[Abs[N[(N[Tanh[N[(N[Log[N[(2.0 * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-1.0 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.20000000000000006e-17

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 12.2

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

   7. Applied rewrites 12.2%

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

   if 5.20000000000000006e-17 < t

   1. Initial program 99.7%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 53.2

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

   6. Applied rewrites 53.2%

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

Alternative 9: 23.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 3.4 \cdot 10^{+209}:\\ \;\;\;\;\left|\tanh \left(\log \left(2 \cdot \frac{eh}{ew}\right) + -1 \cdot \log t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot \left(ew + -0.16666666666666666 \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 3.4e+209)
   (fabs (* (tanh (+ (log (* 2.0 (/ eh ew))) (* -1.0 (log t)))) eh))
   (fabs (* t (+ ew (* -0.16666666666666666 (* ew (* t t))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 3.4e+209) {
		tmp = fabs((tanh((log((2.0 * (eh / ew))) + (-1.0 * log(t)))) * eh));
	} else {
		tmp = fabs((t * (ew + (-0.16666666666666666 * (ew * (t * t))))));
	}
	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 (ew <= 3.4d+209) then
        tmp = abs((tanh((log((2.0d0 * (eh / ew))) + ((-1.0d0) * log(t)))) * eh))
    else
        tmp = abs((t * (ew + ((-0.16666666666666666d0) * (ew * (t * t))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 3.4e+209) {
		tmp = Math.abs((Math.tanh((Math.log((2.0 * (eh / ew))) + (-1.0 * Math.log(t)))) * eh));
	} else {
		tmp = Math.abs((t * (ew + (-0.16666666666666666 * (ew * (t * t))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 3.4e+209:
		tmp = math.fabs((math.tanh((math.log((2.0 * (eh / ew))) + (-1.0 * math.log(t)))) * eh))
	else:
		tmp = math.fabs((t * (ew + (-0.16666666666666666 * (ew * (t * t))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 3.4e+209)
		tmp = abs(Float64(tanh(Float64(log(Float64(2.0 * Float64(eh / ew))) + Float64(-1.0 * log(t)))) * eh));
	else
		tmp = abs(Float64(t * Float64(ew + Float64(-0.16666666666666666 * Float64(ew * Float64(t * t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 3.4e+209)
		tmp = abs((tanh((log((2.0 * (eh / ew))) + (-1.0 * log(t)))) * eh));
	else
		tmp = abs((t * (ew + (-0.16666666666666666 * (ew * (t * t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 3.4e+209], N[Abs[N[(N[Tanh[N[(N[Log[N[(2.0 * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-1.0 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t * N[(ew + N[(-0.16666666666666666 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 3.3999999999999997e209

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 11.3

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

   7. Applied rewrites 11.3%

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

   if 3.3999999999999997e209 < ew

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 79.2

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

   6. Applied rewrites 79.2%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 39.0

         \[\leadsto \left|t \cdot \left(ew + -0.16666666666666666 \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right)\right| \]

   9. Applied rewrites 39.0%

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

Alternative 10: 18.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 1.7 \cdot 10^{-135}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 1.7e-135)
   (fabs (* ew t))
   (fabs (* (tanh (asinh (/ eh (* ew t)))) eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 1.7e-135) {
		tmp = fabs((ew * t));
	} else {
		tmp = fabs((tanh(asinh((eh / (ew * t)))) * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= 1.7e-135:
		tmp = math.fabs((ew * t))
	else:
		tmp = math.fabs((math.tanh(math.asinh((eh / (ew * t)))) * eh))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 1.7e-135)
		tmp = abs(Float64(ew * t));
	else
		tmp = abs(Float64(tanh(asinh(Float64(eh / Float64(ew * t)))) * eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= 1.7e-135)
		tmp = abs((ew * t));
	else
		tmp = abs((tanh(asinh((eh / (ew * t)))) * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 1.7e-135], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 1.69999999999999995e-135

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 48.5

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

   6. Applied rewrites 48.5%

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

   7. Taylor expanded in t around 0

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

      1. lift-*.f64 21.6

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

   9. Applied rewrites 21.6%

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

   if 1.69999999999999995e-135 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 48.2

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

   7. Applied rewrites 48.2%

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

Alternative 11: 13.5% accurate, 47.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 41.5

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

6. Applied rewrites 41.5%

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

7. Taylor expanded in t around 0

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

   1. lift-*.f64 18.5

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

9. Applied rewrites 18.5%

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

Reproduce

herbie shell --seed 2025120 
(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))))))))