rsin B (should all be same)

Percentage Accurate: 76.4% → 99.5%

Time: 6.3s

Alternatives: 22

Speedup: 0.5×

Specification

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r (/ (sin b) (cos (+ a b)))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / cos((a + b)));
}

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(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r * (sin(b) / cos((a + b)))
end function

Java

public static double code(double r, double a, double b) {
	return r * (Math.sin(b) / Math.cos((a + b)));
}

Python

def code(r, a, b):
	return r * (math.sin(b) / math.cos((a + b)))

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / cos(Float64(a + b))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * (sin(b) / cos((a + b)));
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r (/ (sin b) (cos (+ a b)))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / cos((a + b)));
}

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(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r * (sin(b) / cos((a + b)))
end function

Java

public static double code(double r, double a, double b) {
	return r * (Math.sin(b) / Math.cos((a + b)));
}

Python

def code(r, a, b):
	return r * (math.sin(b) / math.cos((a + b)))

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / cos(Float64(a + b))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * (sin(b) / cos((a + b)));
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (fma (cos b) (cos a) (* (- (sin b)) (sin a))))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / fma(cos(b), cos(a), (-sin(b) * sin(a))));
}

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / fma(cos(b), cos(a), Float64(Float64(-sin(b)) * sin(a)))))
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[((-N[Sin[b], $MachinePrecision]) * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

   2. lift-cos.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

   3. +-commutative N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   4. cos-sum N/A

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   5. cos-neg-rev N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

   6. mul-1-neg N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

   7. lower--.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

   8. mul-1-neg N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

   9. cos-neg-rev N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

   10. lower-*.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

   11. lower-cos.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

   12. lower-cos.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

   13. lower-*.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

   14. lift-sin.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

   15. lower-sin.f64 99.5

       \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

3. Applied rewrites 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   2. lift-*.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

   3. lift-cos.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

   4. lift-cos.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

   5. lift-*.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

   6. lift-sin.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

   7. lift-sin.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

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

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

   9. lower-fma.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

   10. lift-cos.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

   11. lift-cos.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

   13. lower-neg.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

   14. lift-sin.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

   15. lift-sin.f64 99.5

       \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

5. Applied rewrites 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
6. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a))))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / ((cos(b) * cos(a)) - (sin(b) * sin(a))));
}

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(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r * (sin(b) / ((cos(b) * cos(a)) - (sin(b) * sin(a))))
end function

Java

public static double code(double r, double a, double b) {
	return r * (Math.sin(b) / ((Math.cos(b) * Math.cos(a)) - (Math.sin(b) * Math.sin(a))));
}

Python

def code(r, a, b):
	return r * (math.sin(b) / ((math.cos(b) * math.cos(a)) - (math.sin(b) * math.sin(a))))

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / Float64(Float64(cos(b) * cos(a)) - Float64(sin(b) * sin(a)))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * (sin(b) / ((cos(b) * cos(a)) - (sin(b) * sin(a))));
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

   2. lift-cos.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

   3. +-commutative N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   4. cos-sum N/A

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   5. cos-neg-rev N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

   6. mul-1-neg N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

   7. lower--.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

   8. mul-1-neg N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

   9. cos-neg-rev N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

   10. lower-*.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

   11. lower-cos.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

   12. lower-cos.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

   13. lower-*.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

   14. lift-sin.f64 N/A

       \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

   15. lower-sin.f64 99.5

       \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

3. Applied rewrites 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
4. Add Preprocessing

Alternative 3: 78.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos b \cdot \cos a\\ t_1 := r \cdot \frac{b}{t\_0 - b \cdot \sin a}\\ \mathbf{if}\;a \leq -4:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.2:\\ \;\;\;\;r \cdot \frac{\sin b}{t\_0 - \sin b \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(a \cdot a\right) - 0.16666666666666666, a \cdot a, 1\right) \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (cos b) (cos a))) (t_1 (* r (/ b (- t_0 (* b (sin a)))))))
   (if (<= a -4.0)
     t_1
     (if (<= a 4.2)
       (*
        r
        (/
         (sin b)
         (-
          t_0
          (*
           (sin b)
           (*
            (fma
             (-
              (*
               (fma (* a a) -0.0001984126984126984 0.008333333333333333)
               (* a a))
              0.16666666666666666)
             (* a a)
             1.0)
            a)))))
       t_1))))

C

double code(double r, double a, double b) {
	double t_0 = cos(b) * cos(a);
	double t_1 = r * (b / (t_0 - (b * sin(a))));
	double tmp;
	if (a <= -4.0) {
		tmp = t_1;
	} else if (a <= 4.2) {
		tmp = r * (sin(b) / (t_0 - (sin(b) * (fma(((fma((a * a), -0.0001984126984126984, 0.008333333333333333) * (a * a)) - 0.16666666666666666), (a * a), 1.0) * a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(cos(b) * cos(a))
	t_1 = Float64(r * Float64(b / Float64(t_0 - Float64(b * sin(a)))))
	tmp = 0.0
	if (a <= -4.0)
		tmp = t_1;
	elseif (a <= 4.2)
		tmp = Float64(r * Float64(sin(b) / Float64(t_0 - Float64(sin(b) * Float64(fma(Float64(Float64(fma(Float64(a * a), -0.0001984126984126984, 0.008333333333333333) * Float64(a * a)) - 0.16666666666666666), Float64(a * a), 1.0) * a)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(r * N[(b / N[(t$95$0 - N[(b * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.0], t$95$1, If[LessEqual[a, 4.2], N[(r * N[(N[Sin[b], $MachinePrecision] / N[(t$95$0 - N[(N[Sin[b], $MachinePrecision] * N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -4 or 4.20000000000000018 < a

   1. Initial program 54.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.3

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.6%

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.6%

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]

   if -4 < a < 4.20000000000000018

   1. Initial program 98.5%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.7

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.7%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\left(a \cdot \left(1 + {a}^{2} \cdot \left({a}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {a}^{2}\right) - \frac{1}{6}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\left(1 + {a}^{2} \cdot \left({a}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {a}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{a}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\left(1 + {a}^{2} \cdot \left({a}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {a}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{a}\right)} \]

   6. Applied rewrites 99.6%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(a \cdot a\right) - 0.16666666666666666, a \cdot a, 1\right) \cdot a\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{b}{\cos b \cdot \cos a - b \cdot \sin a}\\ \mathbf{if}\;a \leq -1.76:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 3.9:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, a \cdot a, 0.041666666666666664\right) \cdot \left(a \cdot a\right) - 0.5, a \cdot a, 1\right) - \sin b \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ b (- (* (cos b) (cos a)) (* b (sin a)))))))
   (if (<= a -1.76)
     t_0
     (if (<= a 3.9)
       (*
        r
        (/
         (sin b)
         (-
          (*
           (cos b)
           (fma
            (-
             (*
              (fma -0.001388888888888889 (* a a) 0.041666666666666664)
              (* a a))
             0.5)
            (* a a)
            1.0))
          (* (sin b) (sin a)))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (b / ((cos(b) * cos(a)) - (b * sin(a))));
	double tmp;
	if (a <= -1.76) {
		tmp = t_0;
	} else if (a <= 3.9) {
		tmp = r * (sin(b) / ((cos(b) * fma(((fma(-0.001388888888888889, (a * a), 0.041666666666666664) * (a * a)) - 0.5), (a * a), 1.0)) - (sin(b) * sin(a))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(b / Float64(Float64(cos(b) * cos(a)) - Float64(b * sin(a)))))
	tmp = 0.0
	if (a <= -1.76)
		tmp = t_0;
	elseif (a <= 3.9)
		tmp = Float64(r * Float64(sin(b) / Float64(Float64(cos(b) * fma(Float64(Float64(fma(-0.001388888888888889, Float64(a * a), 0.041666666666666664) * Float64(a * a)) - 0.5), Float64(a * a), 1.0)) - Float64(sin(b) * sin(a)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(b / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(b * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.76], t$95$0, If[LessEqual[a, 3.9], N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[(N[(N[(N[(-0.001388888888888889 * N[(a * a), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.76000000000000001 or 3.89999999999999991 < a

   1. Initial program 54.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.3

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.6%

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.6%

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]

   if -1.76000000000000001 < a < 3.89999999999999991

   1. Initial program 98.5%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.7

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.7%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\left(1 + {a}^{2} \cdot \left({a}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) - \frac{1}{2}\right)\right)} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \left({a}^{2} \cdot \left({a}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right) - \sin b \cdot \sin a} \]

      2. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \left(\left({a}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) - \frac{1}{2}\right) \cdot {a}^{2} + 1\right) - \sin b \cdot \sin a} \]

      3. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left({a}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) - \frac{1}{2}, \color{blue}{{a}^{2}}, 1\right) - \sin b \cdot \sin a} \]

      4. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left({a}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) - \frac{1}{2}, {\color{blue}{a}}^{2}, 1\right) - \sin b \cdot \sin a} \]

      5. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) \cdot {a}^{2} - \frac{1}{2}, {a}^{2}, 1\right) - \sin b \cdot \sin a} \]

      6. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {a}^{2}\right) \cdot {a}^{2} - \frac{1}{2}, {a}^{2}, 1\right) - \sin b \cdot \sin a} \]

      7. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {a}^{2} + \frac{1}{24}\right) \cdot {a}^{2} - \frac{1}{2}, {a}^{2}, 1\right) - \sin b \cdot \sin a} \]

      8. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {a}^{2}, \frac{1}{24}\right) \cdot {a}^{2} - \frac{1}{2}, {a}^{2}, 1\right) - \sin b \cdot \sin a} \]

      9. unpow2 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, a \cdot a, \frac{1}{24}\right) \cdot {a}^{2} - \frac{1}{2}, {a}^{2}, 1\right) - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, a \cdot a, \frac{1}{24}\right) \cdot {a}^{2} - \frac{1}{2}, {a}^{2}, 1\right) - \sin b \cdot \sin a} \]

      11. unpow2 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, a \cdot a, \frac{1}{24}\right) \cdot \left(a \cdot a\right) - \frac{1}{2}, {a}^{2}, 1\right) - \sin b \cdot \sin a} \]

      12. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, a \cdot a, \frac{1}{24}\right) \cdot \left(a \cdot a\right) - \frac{1}{2}, {a}^{2}, 1\right) - \sin b \cdot \sin a} \]

      13. unpow2 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, a \cdot a, \frac{1}{24}\right) \cdot \left(a \cdot a\right) - \frac{1}{2}, a \cdot \color{blue}{a}, 1\right) - \sin b \cdot \sin a} \]

      14. lower-*.f64 99.5

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, a \cdot a, 0.041666666666666664\right) \cdot \left(a \cdot a\right) - 0.5, a \cdot \color{blue}{a}, 1\right) - \sin b \cdot \sin a} \]

   6. Applied rewrites 99.5%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, a \cdot a, 0.041666666666666664\right) \cdot \left(a \cdot a\right) - 0.5, a \cdot a, 1\right)} - \sin b \cdot \sin a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{b}{\cos b \cdot \cos a - b \cdot \sin a}\\ \mathbf{if}\;a \leq -3.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 3.9:\\ \;\;\;\;r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(a \cdot a\right) - 0.16666666666666666, a \cdot a, 1\right) \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ b (- (* (cos b) (cos a)) (* b (sin a)))))))
   (if (<= a -3.5)
     t_0
     (if (<= a 3.9)
       (*
        r
        (/
         (sin b)
         (fma
          (cos b)
          (cos a)
          (*
           (- (sin b))
           (*
            (fma
             (- (* 0.008333333333333333 (* a a)) 0.16666666666666666)
             (* a a)
             1.0)
            a)))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (b / ((cos(b) * cos(a)) - (b * sin(a))));
	double tmp;
	if (a <= -3.5) {
		tmp = t_0;
	} else if (a <= 3.9) {
		tmp = r * (sin(b) / fma(cos(b), cos(a), (-sin(b) * (fma(((0.008333333333333333 * (a * a)) - 0.16666666666666666), (a * a), 1.0) * a))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(b / Float64(Float64(cos(b) * cos(a)) - Float64(b * sin(a)))))
	tmp = 0.0
	if (a <= -3.5)
		tmp = t_0;
	elseif (a <= 3.9)
		tmp = Float64(r * Float64(sin(b) / fma(cos(b), cos(a), Float64(Float64(-sin(b)) * Float64(fma(Float64(Float64(0.008333333333333333 * Float64(a * a)) - 0.16666666666666666), Float64(a * a), 1.0) * a)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(b / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(b * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.5], t$95$0, If[LessEqual[a, 3.9], N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[((-N[Sin[b], $MachinePrecision]) * N[(N[(N[(N[(0.008333333333333333 * N[(a * a), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.5 or 3.89999999999999991 < a

   1. Initial program 54.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.3

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.6%

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.6%

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]

   if -3.5 < a < 3.89999999999999991

   1. Initial program 98.5%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.7

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.7%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. lift-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      3. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      4. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      5. lift-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      6. lift-sin.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      7. lift-sin.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

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

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a + \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}} \]

      9. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)}} \]

      10. lift-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\color{blue}{\cos b}, \cos a, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      11. lift-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\left(-\sin b\right)} \cdot \sin a\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\color{blue}{\sin b}\right) \cdot \sin a\right)} \]

      15. lift-sin.f64 99.7

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\sin a}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

   6. Taylor expanded in a around 0

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\left(a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}\right)\right)\right)}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \left(\left(1 + {a}^{2} \cdot \left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{a}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \left(\left(1 + {a}^{2} \cdot \left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{a}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \left(\left({a}^{2} \cdot \left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}\right) + 1\right) \cdot a\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \left(\left(\left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}\right) \cdot {a}^{2} + 1\right) \cdot a\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}, {a}^{2}, 1\right) \cdot a\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}, {a}^{2}, 1\right) \cdot a\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}, {a}^{2}, 1\right) \cdot a\right)\right)} \]

      8. unpow2 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(a \cdot a\right) - \frac{1}{6}, {a}^{2}, 1\right) \cdot a\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(a \cdot a\right) - \frac{1}{6}, {a}^{2}, 1\right) \cdot a\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(a \cdot a\right) - \frac{1}{6}, a \cdot a, 1\right) \cdot a\right)\right)} \]

      11. lower-*.f64 99.5

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(a \cdot a\right) - 0.16666666666666666, a \cdot a, 1\right) \cdot a\right)\right)} \]

   8. Applied rewrites 99.5%

      \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \color{blue}{\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(a \cdot a\right) - 0.16666666666666666, a \cdot a, 1\right) \cdot a\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 78.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos b \cdot \cos a\\ t_1 := r \cdot \frac{b}{t\_0 - b \cdot \sin a}\\ \mathbf{if}\;a \leq -3.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.9:\\ \;\;\;\;r \cdot \frac{\sin b}{t\_0 - \sin b \cdot \left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot 0.008333333333333333 - 0.16666666666666666, a \cdot a, 1\right) \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (cos b) (cos a))) (t_1 (* r (/ b (- t_0 (* b (sin a)))))))
   (if (<= a -3.5)
     t_1
     (if (<= a 3.9)
       (*
        r
        (/
         (sin b)
         (-
          t_0
          (*
           (sin b)
           (*
            (fma
             (- (* (* a a) 0.008333333333333333) 0.16666666666666666)
             (* a a)
             1.0)
            a)))))
       t_1))))

C

double code(double r, double a, double b) {
	double t_0 = cos(b) * cos(a);
	double t_1 = r * (b / (t_0 - (b * sin(a))));
	double tmp;
	if (a <= -3.5) {
		tmp = t_1;
	} else if (a <= 3.9) {
		tmp = r * (sin(b) / (t_0 - (sin(b) * (fma((((a * a) * 0.008333333333333333) - 0.16666666666666666), (a * a), 1.0) * a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(cos(b) * cos(a))
	t_1 = Float64(r * Float64(b / Float64(t_0 - Float64(b * sin(a)))))
	tmp = 0.0
	if (a <= -3.5)
		tmp = t_1;
	elseif (a <= 3.9)
		tmp = Float64(r * Float64(sin(b) / Float64(t_0 - Float64(sin(b) * Float64(fma(Float64(Float64(Float64(a * a) * 0.008333333333333333) - 0.16666666666666666), Float64(a * a), 1.0) * a)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(r * N[(b / N[(t$95$0 - N[(b * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.5], t$95$1, If[LessEqual[a, 3.9], N[(r * N[(N[Sin[b], $MachinePrecision] / N[(t$95$0 - N[(N[Sin[b], $MachinePrecision] * N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.5 or 3.89999999999999991 < a

   1. Initial program 54.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.3

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.6%

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.6%

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]

   if -3.5 < a < 3.89999999999999991

   1. Initial program 98.5%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.7

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.7%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\left(a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\left(1 + {a}^{2} \cdot \left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{a}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\left(1 + {a}^{2} \cdot \left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{a}\right)} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\left({a}^{2} \cdot \left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}\right) + 1\right) \cdot a\right)} \]

      4. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\left(\left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}\right) \cdot {a}^{2} + 1\right) \cdot a\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}, {a}^{2}, 1\right) \cdot a\right)} \]

      6. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {a}^{2} - \frac{1}{6}, {a}^{2}, 1\right) \cdot a\right)} \]

      7. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\mathsf{fma}\left({a}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {a}^{2}, 1\right) \cdot a\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\mathsf{fma}\left({a}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {a}^{2}, 1\right) \cdot a\right)} \]

      9. unpow2 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{1}{120} - \frac{1}{6}, {a}^{2}, 1\right) \cdot a\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{1}{120} - \frac{1}{6}, {a}^{2}, 1\right) \cdot a\right)} \]

      11. unpow2 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{1}{120} - \frac{1}{6}, a \cdot a, 1\right) \cdot a\right)} \]

      12. lower-*.f64 99.5

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot 0.008333333333333333 - 0.16666666666666666, a \cdot a, 1\right) \cdot a\right)} \]

   6. Applied rewrites 99.5%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot 0.008333333333333333 - 0.16666666666666666, a \cdot a, 1\right) \cdot a\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{b}{\cos b \cdot \cos a - b \cdot \sin a}\\ \mathbf{if}\;a \leq -0.22:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.35:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(a \cdot a\right) - 0.5, a \cdot a, 1\right) - \sin b \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ b (- (* (cos b) (cos a)) (* b (sin a)))))))
   (if (<= a -0.22)
     t_0
     (if (<= a 0.35)
       (*
        r
        (/
         (sin b)
         (-
          (*
           (cos b)
           (fma (- (* 0.041666666666666664 (* a a)) 0.5) (* a a) 1.0))
          (* (sin b) (sin a)))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (b / ((cos(b) * cos(a)) - (b * sin(a))));
	double tmp;
	if (a <= -0.22) {
		tmp = t_0;
	} else if (a <= 0.35) {
		tmp = r * (sin(b) / ((cos(b) * fma(((0.041666666666666664 * (a * a)) - 0.5), (a * a), 1.0)) - (sin(b) * sin(a))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(b / Float64(Float64(cos(b) * cos(a)) - Float64(b * sin(a)))))
	tmp = 0.0
	if (a <= -0.22)
		tmp = t_0;
	elseif (a <= 0.35)
		tmp = Float64(r * Float64(sin(b) / Float64(Float64(cos(b) * fma(Float64(Float64(0.041666666666666664 * Float64(a * a)) - 0.5), Float64(a * a), 1.0)) - Float64(sin(b) * sin(a)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(b / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(b * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.22], t$95$0, If[LessEqual[a, 0.35], N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(a * a), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.220000000000000001 or 0.34999999999999998 < a

   1. Initial program 54.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.3

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.5%

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.5%

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]

   if -0.220000000000000001 < a < 0.34999999999999998

   1. Initial program 98.6%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.7

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.7%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\left(1 + {a}^{2} \cdot \left(\frac{1}{24} \cdot {a}^{2} - \frac{1}{2}\right)\right)} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \left({a}^{2} \cdot \left(\frac{1}{24} \cdot {a}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right) - \sin b \cdot \sin a} \]

      2. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \left(\left(\frac{1}{24} \cdot {a}^{2} - \frac{1}{2}\right) \cdot {a}^{2} + 1\right) - \sin b \cdot \sin a} \]

      3. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {a}^{2} - \frac{1}{2}, \color{blue}{{a}^{2}}, 1\right) - \sin b \cdot \sin a} \]

      4. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {a}^{2} - \frac{1}{2}, {\color{blue}{a}}^{2}, 1\right) - \sin b \cdot \sin a} \]

      5. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {a}^{2} - \frac{1}{2}, {a}^{2}, 1\right) - \sin b \cdot \sin a} \]

      6. unpow2 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(a \cdot a\right) - \frac{1}{2}, {a}^{2}, 1\right) - \sin b \cdot \sin a} \]

      7. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(a \cdot a\right) - \frac{1}{2}, {a}^{2}, 1\right) - \sin b \cdot \sin a} \]

      8. unpow2 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(a \cdot a\right) - \frac{1}{2}, a \cdot \color{blue}{a}, 1\right) - \sin b \cdot \sin a} \]

      9. lower-*.f64 99.6

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(a \cdot a\right) - 0.5, a \cdot \color{blue}{a}, 1\right) - \sin b \cdot \sin a} \]

   6. Applied rewrites 99.6%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(a \cdot a\right) - 0.5, a \cdot a, 1\right)} - \sin b \cdot \sin a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos b \cdot \cos a\\ t_1 := r \cdot \frac{b}{t\_0 - b \cdot \sin a}\\ \mathbf{if}\;a \leq -1.35:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.1:\\ \;\;\;\;r \cdot \frac{\sin b}{t\_0 - \sin b \cdot \left(\mathsf{fma}\left(-0.16666666666666666, a \cdot a, 1\right) \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (cos b) (cos a))) (t_1 (* r (/ b (- t_0 (* b (sin a)))))))
   (if (<= a -1.35)
     t_1
     (if (<= a 4.1)
       (*
        r
        (/
         (sin b)
         (- t_0 (* (sin b) (* (fma -0.16666666666666666 (* a a) 1.0) a)))))
       t_1))))

C

double code(double r, double a, double b) {
	double t_0 = cos(b) * cos(a);
	double t_1 = r * (b / (t_0 - (b * sin(a))));
	double tmp;
	if (a <= -1.35) {
		tmp = t_1;
	} else if (a <= 4.1) {
		tmp = r * (sin(b) / (t_0 - (sin(b) * (fma(-0.16666666666666666, (a * a), 1.0) * a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(cos(b) * cos(a))
	t_1 = Float64(r * Float64(b / Float64(t_0 - Float64(b * sin(a)))))
	tmp = 0.0
	if (a <= -1.35)
		tmp = t_1;
	elseif (a <= 4.1)
		tmp = Float64(r * Float64(sin(b) / Float64(t_0 - Float64(sin(b) * Float64(fma(-0.16666666666666666, Float64(a * a), 1.0) * a)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(r * N[(b / N[(t$95$0 - N[(b * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.35], t$95$1, If[LessEqual[a, 4.1], N[(r * N[(N[Sin[b], $MachinePrecision] / N[(t$95$0 - N[(N[Sin[b], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3500000000000001 or 4.0999999999999996 < a

   1. Initial program 54.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.3

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.6%

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.6%

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]

   if -1.3500000000000001 < a < 4.0999999999999996

   1. Initial program 98.5%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.7

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.7%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\left(a \cdot \left(1 + \frac{-1}{6} \cdot {a}^{2}\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\left(1 + \frac{-1}{6} \cdot {a}^{2}\right) \cdot \color{blue}{a}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\left(1 + \frac{-1}{6} \cdot {a}^{2}\right) \cdot \color{blue}{a}\right)} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\left(\frac{-1}{6} \cdot {a}^{2} + 1\right) \cdot a\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, {a}^{2}, 1\right) \cdot a\right)} \]

      5. unpow2 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, a \cdot a, 1\right) \cdot a\right)} \]

      6. lower-*.f64 99.5

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \left(\mathsf{fma}\left(-0.16666666666666666, a \cdot a, 1\right) \cdot a\right)} \]

   6. Applied rewrites 99.5%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, a \cdot a, 1\right) \cdot a\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{b}{\cos b \cdot \cos a - b \cdot \sin a}\\ \mathbf{if}\;a \leq -0.16:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.285:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(a \cdot a, -0.5, 1\right) - \sin b \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ b (- (* (cos b) (cos a)) (* b (sin a)))))))
   (if (<= a -0.16)
     t_0
     (if (<= a 0.285)
       (*
        r
        (/ (sin b) (- (* (cos b) (fma (* a a) -0.5 1.0)) (* (sin b) (sin a)))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (b / ((cos(b) * cos(a)) - (b * sin(a))));
	double tmp;
	if (a <= -0.16) {
		tmp = t_0;
	} else if (a <= 0.285) {
		tmp = r * (sin(b) / ((cos(b) * fma((a * a), -0.5, 1.0)) - (sin(b) * sin(a))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(b / Float64(Float64(cos(b) * cos(a)) - Float64(b * sin(a)))))
	tmp = 0.0
	if (a <= -0.16)
		tmp = t_0;
	elseif (a <= 0.285)
		tmp = Float64(r * Float64(sin(b) / Float64(Float64(cos(b) * fma(Float64(a * a), -0.5, 1.0)) - Float64(sin(b) * sin(a)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(b / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(b * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.16], t$95$0, If[LessEqual[a, 0.285], N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.160000000000000003 or 0.284999999999999976 < a

   1. Initial program 54.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.3

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.5%

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.5%

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]

   if -0.160000000000000003 < a < 0.284999999999999976

   1. Initial program 98.6%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.7

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.7%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {a}^{2}\right)} - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \left(\frac{-1}{2} \cdot {a}^{2} + \color{blue}{1}\right) - \sin b \cdot \sin a} \]

      2. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \left({a}^{2} \cdot \frac{-1}{2} + 1\right) - \sin b \cdot \sin a} \]

      3. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left({a}^{2}, \color{blue}{\frac{-1}{2}}, 1\right) - \sin b \cdot \sin a} \]

      4. unpow2 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(a \cdot a, \frac{-1}{2}, 1\right) - \sin b \cdot \sin a} \]

      5. lower-*.f64 99.4

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \mathsf{fma}\left(a \cdot a, -0.5, 1\right) - \sin b \cdot \sin a} \]

   6. Applied rewrites 99.4%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\mathsf{fma}\left(a \cdot a, -0.5, 1\right)} - \sin b \cdot \sin a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 78.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{b}{\cos b \cdot \cos a - b \cdot \sin a}\\ \mathbf{if}\;a \leq -0.16:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.26:\\ \;\;\;\;r \cdot \frac{\sin b}{\mathsf{fma}\left(a, \left(\cos b \cdot a\right) \cdot -0.5 - \sin b, \cos b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ b (- (* (cos b) (cos a)) (* b (sin a)))))))
   (if (<= a -0.16)
     t_0
     (if (<= a 0.26)
       (* r (/ (sin b) (fma a (- (* (* (cos b) a) -0.5) (sin b)) (cos b))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (b / ((cos(b) * cos(a)) - (b * sin(a))));
	double tmp;
	if (a <= -0.16) {
		tmp = t_0;
	} else if (a <= 0.26) {
		tmp = r * (sin(b) / fma(a, (((cos(b) * a) * -0.5) - sin(b)), cos(b)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(b / Float64(Float64(cos(b) * cos(a)) - Float64(b * sin(a)))))
	tmp = 0.0
	if (a <= -0.16)
		tmp = t_0;
	elseif (a <= 0.26)
		tmp = Float64(r * Float64(sin(b) / fma(a, Float64(Float64(Float64(cos(b) * a) * -0.5) - sin(b)), cos(b))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(b / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(b * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.16], t$95$0, If[LessEqual[a, 0.26], N[(r * N[(N[Sin[b], $MachinePrecision] / N[(a * N[(N[(N[(N[Cos[b], $MachinePrecision] * a), $MachinePrecision] * -0.5), $MachinePrecision] - N[Sin[b], $MachinePrecision]), $MachinePrecision] + N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.160000000000000003 or 0.26000000000000001 < a

   1. Initial program 54.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.3

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.5%

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.6%

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]

   if -0.160000000000000003 < a < 0.26000000000000001

   1. Initial program 98.6%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b + a \cdot \left(\frac{-1}{2} \cdot \left(a \cdot \cos b\right) - \sin b\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{a \cdot \left(\frac{-1}{2} \cdot \left(a \cdot \cos b\right) - \sin b\right) + \color{blue}{\cos b}} \]

      2. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(a, \color{blue}{\frac{-1}{2} \cdot \left(a \cdot \cos b\right) - \sin b}, \cos b\right)} \]

      3. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(a, \frac{-1}{2} \cdot \left(a \cdot \cos b\right) - \color{blue}{\sin b}, \cos b\right)} \]

      4. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(a, \left(a \cdot \cos b\right) \cdot \frac{-1}{2} - \sin \color{blue}{b}, \cos b\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(a, \left(a \cdot \cos b\right) \cdot \frac{-1}{2} - \sin \color{blue}{b}, \cos b\right)} \]

      6. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(a, \left(\cos b \cdot a\right) \cdot \frac{-1}{2} - \sin b, \cos b\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(a, \left(\cos b \cdot a\right) \cdot \frac{-1}{2} - \sin b, \cos b\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(a, \left(\cos b \cdot a\right) \cdot \frac{-1}{2} - \sin b, \cos b\right)} \]

      9. lift-sin.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(a, \left(\cos b \cdot a\right) \cdot \frac{-1}{2} - \sin b, \cos b\right)} \]

      10. lower-cos.f64 99.3

          \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(a, \left(\cos b \cdot a\right) \cdot -0.5 - \sin b, \cos b\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(a, \left(\cos b \cdot a\right) \cdot -0.5 - \sin b, \cos b\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 78.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos b \cdot \cos a\\ t_1 := r \cdot \frac{b}{t\_0 - b \cdot \sin a}\\ \mathbf{if}\;a \leq -1.25:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.1:\\ \;\;\;\;r \cdot \frac{\sin b}{t\_0 - \sin b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (cos b) (cos a))) (t_1 (* r (/ b (- t_0 (* b (sin a)))))))
   (if (<= a -1.25)
     t_1
     (if (<= a 1.1) (* r (/ (sin b) (- t_0 (* (sin b) a)))) t_1))))

C

double code(double r, double a, double b) {
	double t_0 = cos(b) * cos(a);
	double t_1 = r * (b / (t_0 - (b * sin(a))));
	double tmp;
	if (a <= -1.25) {
		tmp = t_1;
	} else if (a <= 1.1) {
		tmp = r * (sin(b) / (t_0 - (sin(b) * a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(b) * cos(a)
    t_1 = r * (b / (t_0 - (b * sin(a))))
    if (a <= (-1.25d0)) then
        tmp = t_1
    else if (a <= 1.1d0) then
        tmp = r * (sin(b) / (t_0 - (sin(b) * a)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.cos(b) * Math.cos(a);
	double t_1 = r * (b / (t_0 - (b * Math.sin(a))));
	double tmp;
	if (a <= -1.25) {
		tmp = t_1;
	} else if (a <= 1.1) {
		tmp = r * (Math.sin(b) / (t_0 - (Math.sin(b) * a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.cos(b) * math.cos(a)
	t_1 = r * (b / (t_0 - (b * math.sin(a))))
	tmp = 0
	if a <= -1.25:
		tmp = t_1
	elif a <= 1.1:
		tmp = r * (math.sin(b) / (t_0 - (math.sin(b) * a)))
	else:
		tmp = t_1
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(cos(b) * cos(a))
	t_1 = Float64(r * Float64(b / Float64(t_0 - Float64(b * sin(a)))))
	tmp = 0.0
	if (a <= -1.25)
		tmp = t_1;
	elseif (a <= 1.1)
		tmp = Float64(r * Float64(sin(b) / Float64(t_0 - Float64(sin(b) * a))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = cos(b) * cos(a);
	t_1 = r * (b / (t_0 - (b * sin(a))));
	tmp = 0.0;
	if (a <= -1.25)
		tmp = t_1;
	elseif (a <= 1.1)
		tmp = r * (sin(b) / (t_0 - (sin(b) * a)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(r * N[(b / N[(t$95$0 - N[(b * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.25], t$95$1, If[LessEqual[a, 1.1], N[(r * N[(N[Sin[b], $MachinePrecision] / N[(t$95$0 - N[(N[Sin[b], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.25 or 1.1000000000000001 < a

   1. Initial program 54.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.3

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.6%

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]

   7. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.6%

      \[\leadsto r \cdot \frac{b}{\cos b \cdot \cos a - \color{blue}{b} \cdot \sin a} \]

   if -1.25 < a < 1.1000000000000001

   1. Initial program 98.5%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

      2. lift-cos.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

      4. cos-sum N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      5. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos \left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      6. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(-1 \cdot a\right)} - \sin b \cdot \sin a} \]

      7. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos \left(-1 \cdot a\right) - \sin b \cdot \sin a}} \]

      8. mul-1-neg N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} - \sin b \cdot \sin a} \]

      9. cos-neg-rev N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      10. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a} \]

      11. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b} \cdot \cos a - \sin b \cdot \sin a} \]

      12. lower-cos.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\cos a} - \sin b \cdot \sin a} \]

      13. lower-*.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}} \]

      14. lift-sin.f64 N/A

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\sin b} \cdot \sin a} \]

      15. lower-sin.f64 99.7

          \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\sin a}} \]

   3. Applied rewrites 99.7%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{a}} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.44:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-12}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, b \cdot b, 0.008333333333333333\right) \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.44)
     t_0
     (if (<= b 7.2e-12)
       (*
        r
        (/
         (*
          (fma
           (-
            (*
             (fma -0.0001984126984126984 (* b b) 0.008333333333333333)
             (* b b))
            0.16666666666666666)
           (* b b)
           1.0)
          b)
         (cos (+ a b))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -0.44) {
		tmp = t_0;
	} else if (b <= 7.2e-12) {
		tmp = r * ((fma(((fma(-0.0001984126984126984, (b * b), 0.008333333333333333) * (b * b)) - 0.16666666666666666), (b * b), 1.0) * b) / cos((a + b)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -0.44)
		tmp = t_0;
	elseif (b <= 7.2e-12)
		tmp = Float64(r * Float64(Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(b * b), 0.008333333333333333) * Float64(b * b)) - 0.16666666666666666), Float64(b * b), 1.0) * b) / cos(Float64(a + b))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.44], t$95$0, If[LessEqual[b, 7.2e-12], N[(r * N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(b * b), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.440000000000000002 or 7.2e-12 < b

   1. Initial program 55.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]
   3. Step-by-step derivation

      1. quot-tan N/A

         \[\leadsto r \cdot \tan b \]

      2. lower-tan.f64 54.6

         \[\leadsto r \cdot \tan b \]

   4. Applied rewrites 54.6%

      \[\leadsto r \cdot \color{blue}{\tan b} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{r \cdot \tan b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      3. lower-*.f64 54.6

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      4. cos-sum-rev 54.6

         \[\leadsto \tan b \cdot r \]

      5. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

      6. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -0.440000000000000002 < b < 7.2e-12

   1. Initial program 99.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(a + b\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto r \cdot \frac{\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{b}}{\cos \left(a + b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{b}}{\cos \left(a + b\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, b \cdot b, 0.008333333333333333\right) \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.44:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-12}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot 0.008333333333333333 - 0.16666666666666666, b \cdot b, 1\right) \cdot b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.44)
     t_0
     (if (<= b 7.2e-12)
       (*
        r
        (/
         (*
          (fma
           (- (* (* b b) 0.008333333333333333) 0.16666666666666666)
           (* b b)
           1.0)
          b)
         (cos (* (+ (/ b a) 1.0) a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -0.44) {
		tmp = t_0;
	} else if (b <= 7.2e-12) {
		tmp = r * ((fma((((b * b) * 0.008333333333333333) - 0.16666666666666666), (b * b), 1.0) * b) / cos((((b / a) + 1.0) * a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -0.44)
		tmp = t_0;
	elseif (b <= 7.2e-12)
		tmp = Float64(r * Float64(Float64(fma(Float64(Float64(Float64(b * b) * 0.008333333333333333) - 0.16666666666666666), Float64(b * b), 1.0) * b) / cos(Float64(Float64(Float64(b / a) + 1.0) * a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.44], t$95$0, If[LessEqual[b, 7.2e-12], N[(r * N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(N[(N[(b / a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.440000000000000002 or 7.2e-12 < b

   1. Initial program 55.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]
   3. Step-by-step derivation

      1. quot-tan N/A

         \[\leadsto r \cdot \tan b \]

      2. lower-tan.f64 54.6

         \[\leadsto r \cdot \tan b \]

   4. Applied rewrites 54.6%

      \[\leadsto r \cdot \color{blue}{\tan b} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{r \cdot \tan b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      3. lower-*.f64 54.6

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      4. cos-sum-rev 54.6

         \[\leadsto \tan b \cdot r \]

      5. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

      6. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -0.440000000000000002 < b < 7.2e-12

   1. Initial program 99.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around inf

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a \cdot \left(1 + \frac{b}{a}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \left(\left(1 + \frac{b}{a}\right) \cdot \color{blue}{a}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \left(\left(1 + \frac{b}{a}\right) \cdot \color{blue}{a}\right)} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto r \cdot \frac{\sin b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

      5. lower-/.f64 99.3

         \[\leadsto r \cdot \frac{\sin b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(\left(\frac{b}{a} + 1\right) \cdot a\right)}} \]

   5. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)}}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto r \cdot \frac{\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

      4. *-commutative N/A

         \[\leadsto r \cdot \frac{\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

      6. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

      7. *-commutative N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left({b}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

      9. pow2 N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{1}{120} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{1}{120} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

      11. pow2 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{1}{120} - \frac{1}{6}, b \cdot b, 1\right) \cdot b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

      12. lift-*.f64 99.3

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot 0.008333333333333333 - 0.16666666666666666, b \cdot b, 1\right) \cdot b}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]

   7. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(b \cdot b\right) \cdot 0.008333333333333333 - 0.16666666666666666, b \cdot b, 1\right) \cdot b}}{\cos \left(\left(\frac{b}{a} + 1\right) \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 76.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.44:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-12}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(0.008333333333333333 \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.44)
     t_0
     (if (<= b 7.2e-12)
       (*
        r
        (/
         (*
          (fma
           (- (* 0.008333333333333333 (* b b)) 0.16666666666666666)
           (* b b)
           1.0)
          b)
         (cos (+ a b))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -0.44) {
		tmp = t_0;
	} else if (b <= 7.2e-12) {
		tmp = r * ((fma(((0.008333333333333333 * (b * b)) - 0.16666666666666666), (b * b), 1.0) * b) / cos((a + b)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -0.44)
		tmp = t_0;
	elseif (b <= 7.2e-12)
		tmp = Float64(r * Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(b * b)) - 0.16666666666666666), Float64(b * b), 1.0) * b) / cos(Float64(a + b))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.44], t$95$0, If[LessEqual[b, 7.2e-12], N[(r * N[(N[(N[(N[(N[(0.008333333333333333 * N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.440000000000000002 or 7.2e-12 < b

   1. Initial program 55.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]
   3. Step-by-step derivation

      1. quot-tan N/A

         \[\leadsto r \cdot \tan b \]

      2. lower-tan.f64 54.6

         \[\leadsto r \cdot \tan b \]

   4. Applied rewrites 54.6%

      \[\leadsto r \cdot \color{blue}{\tan b} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{r \cdot \tan b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      3. lower-*.f64 54.6

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      4. cos-sum-rev 54.6

         \[\leadsto \tan b \cdot r \]

      5. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

      6. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -0.440000000000000002 < b < 7.2e-12

   1. Initial program 99.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)}}{\cos \left(a + b\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto r \cdot \frac{\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}}{\cos \left(a + b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{b}}{\cos \left(a + b\right)} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) + 1\right) \cdot b}{\cos \left(a + b\right)} \]

      4. *-commutative N/A

         \[\leadsto r \cdot \frac{\left(\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right) \cdot {b}^{2} + 1\right) \cdot b}{\cos \left(a + b\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      6. lower--.f64 N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}, {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      8. unpow2 N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot \left(b \cdot b\right) - \frac{1}{6}, {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot \left(b \cdot b\right) - \frac{1}{6}, {b}^{2}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      10. unpow2 N/A

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot \left(b \cdot b\right) - \frac{1}{6}, b \cdot b, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      11. lower-*.f64 99.3

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(0.008333333333333333 \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b}{\cos \left(a + b\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 76.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.44:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{r \cdot \left(\mathsf{fma}\left(b \cdot b, \left(0.008333333333333333 \cdot b\right) \cdot b - 0.16666666666666666, 1\right) \cdot b\right)}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.44)
     t_0
     (if (<= b 7.2e-12)
       (/
        (*
         r
         (*
          (fma
           (* b b)
           (- (* (* 0.008333333333333333 b) b) 0.16666666666666666)
           1.0)
          b))
        (cos (+ a b)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -0.44) {
		tmp = t_0;
	} else if (b <= 7.2e-12) {
		tmp = (r * (fma((b * b), (((0.008333333333333333 * b) * b) - 0.16666666666666666), 1.0) * b)) / cos((a + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -0.44)
		tmp = t_0;
	elseif (b <= 7.2e-12)
		tmp = Float64(Float64(r * Float64(fma(Float64(b * b), Float64(Float64(Float64(0.008333333333333333 * b) * b) - 0.16666666666666666), 1.0) * b)) / cos(Float64(a + b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.44], t$95$0, If[LessEqual[b, 7.2e-12], N[(N[(r * N[(N[(N[(b * b), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * b), $MachinePrecision] * b), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.440000000000000002 or 7.2e-12 < b

   1. Initial program 55.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]
   3. Step-by-step derivation

      1. quot-tan N/A

         \[\leadsto r \cdot \tan b \]

      2. lower-tan.f64 54.6

         \[\leadsto r \cdot \tan b \]

   4. Applied rewrites 54.6%

      \[\leadsto r \cdot \color{blue}{\tan b} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{r \cdot \tan b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      3. lower-*.f64 54.6

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      4. cos-sum-rev 54.6

         \[\leadsto \tan b \cdot r \]

      5. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

      6. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -0.440000000000000002 < b < 7.2e-12

   1. Initial program 99.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(a + b\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto r \cdot \frac{\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{b}}{\cos \left(a + b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{b}}{\cos \left(a + b\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, b \cdot b, 0.008333333333333333\right) \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b}}{\cos \left(a + b\right)} \]

   5. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot \left(b \cdot b\right) - \frac{1}{6}, b \cdot b, 1\right) \cdot b}{\cos \left(a + b\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

      \[\leadsto r \cdot \frac{\mathsf{fma}\left(0.008333333333333333 \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b}{\cos \left(a + b\right)} \]

   9. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{r \cdot \left(\mathsf{fma}\left(b \cdot b, \left(0.008333333333333333 \cdot b\right) \cdot b - 0.16666666666666666, 1\right) \cdot b\right)}{\cos \left(a + b\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 76.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.44:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-12}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.44)
     t_0
     (if (<= b 7.2e-12)
       (* r (/ (* (fma (* b b) -0.16666666666666666 1.0) b) (cos (+ a b))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -0.44) {
		tmp = t_0;
	} else if (b <= 7.2e-12) {
		tmp = r * ((fma((b * b), -0.16666666666666666, 1.0) * b) / cos((a + b)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -0.44)
		tmp = t_0;
	elseif (b <= 7.2e-12)
		tmp = Float64(r * Float64(Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * b) / cos(Float64(a + b))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.44], t$95$0, If[LessEqual[b, 7.2e-12], N[(r * N[(N[(N[(N[(b * b), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.440000000000000002 or 7.2e-12 < b

   1. Initial program 55.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]
   3. Step-by-step derivation

      1. quot-tan N/A

         \[\leadsto r \cdot \tan b \]

      2. lower-tan.f64 54.6

         \[\leadsto r \cdot \tan b \]

   4. Applied rewrites 54.6%

      \[\leadsto r \cdot \color{blue}{\tan b} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{r \cdot \tan b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      3. lower-*.f64 54.6

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      4. cos-sum-rev 54.6

         \[\leadsto \tan b \cdot r \]

      5. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

      6. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -0.440000000000000002 < b < 7.2e-12

   1. Initial program 99.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)}}{\cos \left(a + b\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto r \cdot \frac{\left(1 + \frac{-1}{6} \cdot {b}^{2}\right) \cdot \color{blue}{b}}{\cos \left(a + b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto r \cdot \frac{\left(1 + \frac{-1}{6} \cdot {b}^{2}\right) \cdot \color{blue}{b}}{\cos \left(a + b\right)} \]

      3. +-commutative N/A

         \[\leadsto r \cdot \frac{\left(\frac{-1}{6} \cdot {b}^{2} + 1\right) \cdot b}{\cos \left(a + b\right)} \]

      4. *-commutative N/A

         \[\leadsto r \cdot \frac{\left({b}^{2} \cdot \frac{-1}{6} + 1\right) \cdot b}{\cos \left(a + b\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left({b}^{2}, \frac{-1}{6}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      6. unpow2 N/A

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(b \cdot b, \frac{-1}{6}, 1\right) \cdot b}{\cos \left(a + b\right)} \]

      7. lower-*.f64 99.2

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b}{\cos \left(a + b\right)} \]

   4. Applied rewrites 99.2%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 76.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.44:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-12}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.44) t_0 (if (<= b 7.2e-12) (* r (/ b (cos (+ a b)))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -0.44) {
		tmp = t_0;
	} else if (b <= 7.2e-12) {
		tmp = r * (b / cos((a + b)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(b) * r
    if (b <= (-0.44d0)) then
        tmp = t_0
    else if (b <= 7.2d-12) then
        tmp = r * (b / cos((a + b)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.tan(b) * r;
	double tmp;
	if (b <= -0.44) {
		tmp = t_0;
	} else if (b <= 7.2e-12) {
		tmp = r * (b / Math.cos((a + b)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.tan(b) * r
	tmp = 0
	if b <= -0.44:
		tmp = t_0
	elif b <= 7.2e-12:
		tmp = r * (b / math.cos((a + b)))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -0.44)
		tmp = t_0;
	elseif (b <= 7.2e-12)
		tmp = Float64(r * Float64(b / cos(Float64(a + b))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = tan(b) * r;
	tmp = 0.0;
	if (b <= -0.44)
		tmp = t_0;
	elseif (b <= 7.2e-12)
		tmp = r * (b / cos((a + b)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.44], t$95$0, If[LessEqual[b, 7.2e-12], N[(r * N[(b / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.440000000000000002 or 7.2e-12 < b

   1. Initial program 55.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]
   3. Step-by-step derivation

      1. quot-tan N/A

         \[\leadsto r \cdot \tan b \]

      2. lower-tan.f64 54.6

         \[\leadsto r \cdot \tan b \]

   4. Applied rewrites 54.6%

      \[\leadsto r \cdot \color{blue}{\tan b} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{r \cdot \tan b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      3. lower-*.f64 54.6

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      4. cos-sum-rev 54.6

         \[\leadsto \tan b \cdot r \]

      5. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

      6. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -0.440000000000000002 < b < 7.2e-12

   1. Initial program 99.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos \left(a + b\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.1%

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 76.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.44:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-12}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.44) t_0 (if (<= b 7.2e-12) (* r (/ b (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -0.44) {
		tmp = t_0;
	} else if (b <= 7.2e-12) {
		tmp = r * (b / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(b) * r
    if (b <= (-0.44d0)) then
        tmp = t_0
    else if (b <= 7.2d-12) then
        tmp = r * (b / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.tan(b) * r;
	double tmp;
	if (b <= -0.44) {
		tmp = t_0;
	} else if (b <= 7.2e-12) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.tan(b) * r
	tmp = 0
	if b <= -0.44:
		tmp = t_0
	elif b <= 7.2e-12:
		tmp = r * (b / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -0.44)
		tmp = t_0;
	elseif (b <= 7.2e-12)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = tan(b) * r;
	tmp = 0.0;
	if (b <= -0.44)
		tmp = t_0;
	elseif (b <= 7.2e-12)
		tmp = r * (b / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.44], t$95$0, If[LessEqual[b, 7.2e-12], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.440000000000000002 or 7.2e-12 < b

   1. Initial program 55.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]
   3. Step-by-step derivation

      1. quot-tan N/A

         \[\leadsto r \cdot \tan b \]

      2. lower-tan.f64 54.6

         \[\leadsto r \cdot \tan b \]

   4. Applied rewrites 54.6%

      \[\leadsto r \cdot \color{blue}{\tan b} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{r \cdot \tan b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      3. lower-*.f64 54.6

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      4. cos-sum-rev 54.6

         \[\leadsto \tan b \cdot r \]

      5. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

      6. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -0.440000000000000002 < b < 7.2e-12

   1. Initial program 99.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in b around 0

      \[\leadsto r \cdot \color{blue}{\frac{b}{\cos a}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto r \cdot \frac{b}{\color{blue}{\cos a}} \]

      2. lower-cos.f64 99.1

         \[\leadsto r \cdot \frac{b}{\cos a} \]

   4. Applied rewrites 99.1%

      \[\leadsto r \cdot \color{blue}{\frac{b}{\cos a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 76.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -0.44:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-12}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -0.44) t_0 (if (<= b 7.2e-12) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -0.44) {
		tmp = t_0;
	} else if (b <= 7.2e-12) {
		tmp = b * (r / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(b) * r
    if (b <= (-0.44d0)) then
        tmp = t_0
    else if (b <= 7.2d-12) then
        tmp = b * (r / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.tan(b) * r;
	double tmp;
	if (b <= -0.44) {
		tmp = t_0;
	} else if (b <= 7.2e-12) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.tan(b) * r
	tmp = 0
	if b <= -0.44:
		tmp = t_0
	elif b <= 7.2e-12:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -0.44)
		tmp = t_0;
	elseif (b <= 7.2e-12)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = tan(b) * r;
	tmp = 0.0;
	if (b <= -0.44)
		tmp = t_0;
	elseif (b <= 7.2e-12)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -0.44], t$95$0, If[LessEqual[b, 7.2e-12], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.440000000000000002 or 7.2e-12 < b

   1. Initial program 55.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]
   3. Step-by-step derivation

      1. quot-tan N/A

         \[\leadsto r \cdot \tan b \]

      2. lower-tan.f64 54.6

         \[\leadsto r \cdot \tan b \]

   4. Applied rewrites 54.6%

      \[\leadsto r \cdot \color{blue}{\tan b} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{r \cdot \tan b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      3. lower-*.f64 54.6

         \[\leadsto \color{blue}{\tan b \cdot r} \]

      4. cos-sum-rev 54.6

         \[\leadsto \tan b \cdot r \]

      5. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

      6. *-commutative 54.6

         \[\leadsto \tan b \cdot r \]

   6. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   if -0.440000000000000002 < b < 7.2e-12

   1. Initial program 99.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto b \cdot \color{blue}{\frac{r}{\cos a}} \]

      2. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\frac{r}{\cos a}} \]

      3. lower-/.f64 N/A

         \[\leadsto b \cdot \frac{r}{\color{blue}{\cos a}} \]

      4. lower-cos.f64 99.1

         \[\leadsto b \cdot \frac{r}{\cos a} \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{b \cdot \frac{r}{\cos a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin b \cdot r}{\cos \left(a + b\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (/ (* (sin b) r) (cos (+ a b))))

C

double code(double r, double a, double b) {
	return (sin(b) * r) / cos((a + b));
}

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(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (sin(b) * r) / cos((a + b))
end function

Java

public static double code(double r, double a, double b) {
	return (Math.sin(b) * r) / Math.cos((a + b));
}

Python

def code(r, a, b):
	return (math.sin(b) * r) / math.cos((a + b))

Julia

function code(r, a, b)
	return Float64(Float64(sin(b) * r) / cos(Float64(a + b)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (sin(b) * r) / cos((a + b));
end

Wolfram

code[r_, a_, b_] := N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(a + b\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos \left(a + b\right)}} \]

   3. lift-sin.f64 N/A

      \[\leadsto r \cdot \frac{\color{blue}{\sin b}}{\cos \left(a + b\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a + b\right)}} \]

   5. lift-cos.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos \left(a + b\right)}} \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(a + b\right)}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(a + b\right)}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos \left(a + b\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos \left(a + b\right)} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{\color{blue}{\sin b} \cdot r}{\cos \left(a + b\right)} \]

   11. lift-cos.f64 N/A

       \[\leadsto \frac{\sin b \cdot r}{\color{blue}{\cos \left(a + b\right)}} \]

   12. lift-+.f64 76.4

       \[\leadsto \frac{\sin b \cdot r}{\cos \color{blue}{\left(a + b\right)}} \]

3. Applied rewrites 76.4%

   \[\leadsto \color{blue}{\frac{\sin b \cdot r}{\cos \left(a + b\right)}} \]
4. Add Preprocessing

Alternative 21: 60.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan b \cdot r \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* (tan b) r))

C

double code(double r, double a, double b) {
	return tan(b) * r;
}

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(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = tan(b) * r
end function

Java

public static double code(double r, double a, double b) {
	return Math.tan(b) * r;
}

Python

def code(r, a, b):
	return math.tan(b) * r

Julia

function code(r, a, b)
	return Float64(tan(b) * r)
end

MATLAB

function tmp = code(r, a, b)
	tmp = tan(b) * r;
end

Wolfram

code[r_, a_, b_] := N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

2. Taylor expanded in a around 0

   \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]
3. Step-by-step derivation

   1. quot-tan N/A

      \[\leadsto r \cdot \tan b \]

   2. lower-tan.f64 60.3

      \[\leadsto r \cdot \tan b \]

4. Applied rewrites 60.3%

   \[\leadsto r \cdot \color{blue}{\tan b} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{r \cdot \tan b} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   3. lower-*.f64 60.3

      \[\leadsto \color{blue}{\tan b \cdot r} \]

   4. cos-sum-rev 60.3

      \[\leadsto \tan b \cdot r \]

   5. *-commutative 60.3

      \[\leadsto \tan b \cdot r \]

   6. *-commutative 60.3

      \[\leadsto \tan b \cdot r \]

6. Applied rewrites 60.3%

   \[\leadsto \color{blue}{\tan b \cdot r} \]
7. Add Preprocessing

Alternative 22: 34.2% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ r \cdot b \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r b))

C

double code(double r, double a, double b) {
	return r * b;
}

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(r, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r * b
end function

Java

public static double code(double r, double a, double b) {
	return r * b;
}

Python

def code(r, a, b):
	return r * b

Julia

function code(r, a, b)
	return Float64(r * b)
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * b;
end

Wolfram

code[r_, a_, b_] := N[(r * b), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

2. Taylor expanded in a around 0

   \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\cos b}} \]
3. Step-by-step derivation

   1. quot-tan N/A

      \[\leadsto r \cdot \tan b \]

   2. lower-tan.f64 60.3

      \[\leadsto r \cdot \tan b \]

4. Applied rewrites 60.3%

   \[\leadsto r \cdot \color{blue}{\tan b} \]

5. Taylor expanded in b around 0

   \[\leadsto r \cdot b \]
6. Step-by-step derivation

7. Applied rewrites 34.2%

   \[\leadsto r \cdot b \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025117 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))