Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 15.1s

Alternatives: 5

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

Alternative 2: 98.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 90.3

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

4. Applied rewrites 90.3%

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

5. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 90.3

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

7. Applied rewrites 90.3%

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

9. Applied rewrites 90.1%

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

10. Taylor expanded in eh around 0

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

12. Applied rewrites 98.2%

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

Alternative 3: 75.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left(-eh\right) \cdot \left|t\right|}{ew}\\ t_2 := \cos \left(\left|t\right|\right)\\ \mathbf{if}\;\left|t\right| \leq 2.7 \cdot 10^{+31}:\\ \;\;\;\;\left|\frac{t\_2 \cdot ew}{\sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}} - \tanh \sinh^{-1} t\_1 \cdot \left(\left|t\right| \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\_2\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (* (- eh) (fabs t)) ew)) (t_2 (cos (fabs t))))
   (if (<= (fabs t) 2.7e+31)
     (fabs
      (-
       (/ (* t_2 ew) (sqrt (fma t_1 t_1 1.0)))
       (* (tanh (asinh t_1)) (* (fabs t) eh))))
     (fabs (* ew t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = (-eh * fabs(t)) / ew;
	double t_2 = cos(fabs(t));
	double tmp;
	if (fabs(t) <= 2.7e+31) {
		tmp = fabs((((t_2 * ew) / sqrt(fma(t_1, t_1, 1.0))) - (tanh(asinh(t_1)) * (fabs(t) * eh))));
	} else {
		tmp = fabs((ew * t_2));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(Float64(-eh) * abs(t)) / ew)
	t_2 = cos(abs(t))
	tmp = 0.0
	if (abs(t) <= 2.7e+31)
		tmp = abs(Float64(Float64(Float64(t_2 * ew) / sqrt(fma(t_1, t_1, 1.0))) - Float64(tanh(asinh(t_1)) * Float64(abs(t) * eh))));
	else
		tmp = abs(Float64(ew * t_2));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[((-eh) * N[Abs[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Abs[t], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[t], $MachinePrecision], 2.7e+31], N[Abs[N[(N[(N[(t$95$2 * ew), $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * t$95$2), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.69999999999999986e31

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 90.3

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

   7. Applied rewrites 90.3%

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

   9. Applied rewrites 90.1%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 66.2%

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

   if 2.69999999999999986e31 < t

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 61.7

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

   6. Applied rewrites 61.7%

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

Alternative 4: 61.7% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 61.7

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

6. Applied rewrites 61.7%

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

Alternative 5: 42.2% accurate, 25.0× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left|ew\right|}\\ t\_1 \cdot t\_1 \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sqrt (fabs ew)))) (* t_1 t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = sqrt(fabs(ew));
	return t_1 * 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 = sqrt(abs(ew))
    code = t_1 * t_1
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sqrt(Math.abs(ew));
	return t_1 * t_1;
}

Python

def code(eh, ew, t):
	t_1 = math.sqrt(math.fabs(ew))
	return t_1 * t_1

Julia

function code(eh, ew, t)
	t_1 = sqrt(abs(ew))
	return Float64(t_1 * t_1)
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = sqrt(abs(ew));
	tmp = t_1 * t_1;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sqrt[N[Abs[ew], $MachinePrecision]], $MachinePrecision]}, N[(t$95$1 * t$95$1), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 41.0%

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

4. Taylor expanded in t around 0

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

   1. lower-sqrt.f64 20.1

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

6. Applied rewrites 20.1%

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

7. Taylor expanded in t around 0

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

   1. lower-sqrt.f64 21.6

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

9. Applied rewrites 21.6%

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

Reproduce

herbie shell --seed 2025171 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))