rsin B (should all be same)

Percentage Accurate: 77.1% → 99.5%

Time: 5.2s

Alternatives: 18

Speedup: 1.0×

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: 77.1% 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

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

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 78.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Add Preprocessing
3. 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

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

4. Applied rewrites 99.5%

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

Alternative 2: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.095:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{elif}\;b \leq 2.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{r \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right) \cdot b\right) \cdot b - 0.16666666666666666, b \cdot b, 1\right) \cdot b\right)}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.095)
   (/ (* (sin b) r) (cos b))
   (if (<= b 2.35e-7)
     (/
      (*
       r
       (*
        (fma
         (-
          (* (* (fma (* b b) -0.0001984126984126984 0.008333333333333333) b) b)
          0.16666666666666666)
         (* b b)
         1.0)
        b))
      (cos (+ a b)))
     (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.095) {
		tmp = (sin(b) * r) / cos(b);
	} else if (b <= 2.35e-7) {
		tmp = (r * (fma((((fma((b * b), -0.0001984126984126984, 0.008333333333333333) * b) * b) - 0.16666666666666666), (b * b), 1.0) * b)) / cos((a + b));
	} else {
		tmp = r * tan(b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.095)
		tmp = Float64(Float64(sin(b) * r) / cos(b));
	elseif (b <= 2.35e-7)
		tmp = Float64(Float64(r * Float64(fma(Float64(Float64(Float64(fma(Float64(b * b), -0.0001984126984126984, 0.008333333333333333) * b) * b) - 0.16666666666666666), Float64(b * b), 1.0) * b)) / cos(Float64(a + b)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.095], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.35e-7], N[(N[(r * N[(N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.095000000000000001

   1. Initial program 49.7%

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

   3. Taylor expanded in a around inf

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a \cdot \left(1 + \frac{b}{a}\right)\right)}} \]
   4. 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 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 12.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 12.7

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

   10. Applied rewrites 12.7%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 50.6%

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

   if -0.095000000000000001 < b < 2.35e-7

   1. Initial program 98.9%

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

   3. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(a + b\right)} \]
   4. 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

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

   5. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 98.9%

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

   if 2.35e-7 < b

   1. Initial program 60.2%

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

   3. Taylor expanded in a around 0

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

      1. quot-tan N/A

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

      2. lower-tan.f64 61.1

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

   5. Applied rewrites 61.1%

      \[\leadsto r \cdot \color{blue}{\tan b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.095:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{elif}\;b \leq 2.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{r \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right) \cdot b\right) \cdot b - 0.16666666666666666, b \cdot b, 1\right) \cdot b\right)}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.095:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 2.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{r \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right) \cdot b\right) \cdot b - 0.16666666666666666, b \cdot b, 1\right) \cdot b\right)}{\cos \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.095)
   (* r (/ (sin b) (cos b)))
   (if (<= b 2.35e-7)
     (/
      (*
       r
       (*
        (fma
         (-
          (* (* (fma (* b b) -0.0001984126984126984 0.008333333333333333) b) b)
          0.16666666666666666)
         (* b b)
         1.0)
        b))
      (cos (+ a b)))
     (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.095) {
		tmp = r * (sin(b) / cos(b));
	} else if (b <= 2.35e-7) {
		tmp = (r * (fma((((fma((b * b), -0.0001984126984126984, 0.008333333333333333) * b) * b) - 0.16666666666666666), (b * b), 1.0) * b)) / cos((a + b));
	} else {
		tmp = r * tan(b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.095)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	elseif (b <= 2.35e-7)
		tmp = Float64(Float64(r * Float64(fma(Float64(Float64(Float64(fma(Float64(b * b), -0.0001984126984126984, 0.008333333333333333) * b) * b) - 0.16666666666666666), Float64(b * b), 1.0) * b)) / cos(Float64(a + b)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.095], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.35e-7], N[(N[(r * N[(N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.095000000000000001

   1. Initial program 49.7%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 50.5%

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

   if -0.095000000000000001 < b < 2.35e-7

   1. Initial program 98.9%

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

   3. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(a + b\right)} \]
   4. 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

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

   5. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 98.9%

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

   if 2.35e-7 < b

   1. Initial program 60.2%

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

   3. Taylor expanded in a around 0

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

      1. quot-tan N/A

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

      2. lower-tan.f64 61.1

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

   5. Applied rewrites 61.1%

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

Alternative 4: 77.1% 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 78.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Add Preprocessing
3. 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 78.5

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

4. Applied rewrites 78.5%

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

Alternative 5: 77.1% accurate, 1.0× speedup

Math

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

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(sin(b) * Float64(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[Sin[b], $MachinePrecision] * N[(r / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Add Preprocessing
3. 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

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

4. Applied rewrites 99.5%

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

   1. lift-cos.f64 N/A

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

   2. unpow1 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{{\cos a}^{1}} - \sin b \cdot \sin a} \]

   3. metadata-eval N/A

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

   4. pow-neg N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\frac{1}{{\cos a}^{-1}}} - \sin b \cdot \sin a} \]

   5. inv-pow N/A

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

   6. lower-/.f64 N/A

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

   7. inv-pow N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \frac{1}{\color{blue}{{\cos a}^{-1}}} - \sin b \cdot \sin a} \]

   8. lower-pow.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \frac{1}{\color{blue}{{\cos a}^{-1}}} - \sin b \cdot \sin a} \]

   9. lift-cos.f64 99.5

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \frac{1}{{\color{blue}{\cos a}}^{-1}} - \sin b \cdot \sin a} \]

6. Applied rewrites 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \color{blue}{\frac{1}{{\cos a}^{-1}}} - \sin b \cdot \sin a} \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \frac{1}{{\color{blue}{\cos a}}^{-1}} - \sin b \cdot \sin a} \]

   2. lift-pow.f64 N/A

      \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \frac{1}{\color{blue}{{\cos a}^{-1}}} - \sin b \cdot \sin a} \]

   3. inv-pow N/A

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

   4. lower-/.f64 N/A

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

   5. lift-cos.f64 99.5

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

8. Applied rewrites 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \frac{1}{\color{blue}{\frac{1}{\cos a}}} - \sin b \cdot \sin a} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. associate-*r/ N/A

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

10. Applied rewrites 78.4%

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

Alternative 6: 77.1% 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]

Derivation

1. Initial program 78.4%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 7: 76.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.095 \lor \neg \left(b \leq 2.35 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right) \cdot b\right) \cdot b - 0.16666666666666666, b \cdot b, 1\right) \cdot b\right)}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.095) (not (<= b 2.35e-7)))
   (* r (tan b))
   (/
    (*
     r
     (*
      (fma
       (-
        (* (* (fma (* b b) -0.0001984126984126984 0.008333333333333333) b) b)
        0.16666666666666666)
       (* b b)
       1.0)
      b))
    (cos (+ a b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.095) || !(b <= 2.35e-7)) {
		tmp = r * tan(b);
	} else {
		tmp = (r * (fma((((fma((b * b), -0.0001984126984126984, 0.008333333333333333) * b) * b) - 0.16666666666666666), (b * b), 1.0) * b)) / cos((a + b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.095) || !(b <= 2.35e-7))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(Float64(r * Float64(fma(Float64(Float64(Float64(fma(Float64(b * b), -0.0001984126984126984, 0.008333333333333333) * b) * b) - 0.16666666666666666), Float64(b * b), 1.0) * b)) / cos(Float64(a + b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.095], N[Not[LessEqual[b, 2.35e-7]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * N[(N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.095000000000000001 or 2.35e-7 < b

   1. Initial program 55.2%

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

   3. Taylor expanded in a around 0

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

      1. quot-tan N/A

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

      2. lower-tan.f64 56.1

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

   5. Applied rewrites 56.1%

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

   if -0.095000000000000001 < b < 2.35e-7

   1. Initial program 98.9%

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

   3. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(a + b\right)} \]
   4. 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

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

   5. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 98.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.095 \lor \neg \left(b \leq 2.35 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right) \cdot b\right) \cdot b - 0.16666666666666666, b \cdot b, 1\right) \cdot b\right)}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.095 \lor \neg \left(b \leq 2.35 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, b \cdot b, 0.008333333333333333\right) \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.095) (not (<= b 2.35e-7)))
   (* r (tan b))
   (*
    r
    (/
     (*
      (fma
       (-
        (* (fma -0.0001984126984126984 (* b b) 0.008333333333333333) (* b b))
        0.16666666666666666)
       (* b b)
       1.0)
      b)
     (cos (+ a b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.095) || !(b <= 2.35e-7)) {
		tmp = r * tan(b);
	} else {
		tmp = r * ((fma(((fma(-0.0001984126984126984, (b * b), 0.008333333333333333) * (b * b)) - 0.16666666666666666), (b * b), 1.0) * b) / cos((a + b)));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.095) || !(b <= 2.35e-7))
		tmp = Float64(r * tan(b));
	else
		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))));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.095], N[Not[LessEqual[b, 2.35e-7]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if b < -0.095000000000000001 or 2.35e-7 < b

   1. Initial program 55.2%

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

   3. Taylor expanded in a around 0

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

      1. quot-tan N/A

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

      2. lower-tan.f64 56.1

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

   5. Applied rewrites 56.1%

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

   if -0.095000000000000001 < b < 2.35e-7

   1. Initial program 98.9%

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

   3. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(a + b\right)} \]
   4. 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

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

   5. Applied rewrites 98.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.095 \lor \neg \left(b \leq 2.35 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, b \cdot b, 0.008333333333333333\right) \cdot \left(b \cdot b\right) - 0.16666666666666666, b \cdot b, 1\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0082 \lor \neg \left(b \leq 2.35 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.0082) (not (<= b 2.35e-7)))
   (* r (tan b))
   (* r (/ (* (fma (* b b) -0.16666666666666666 1.0) b) (cos (+ a b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0082) || !(b <= 2.35e-7)) {
		tmp = r * tan(b);
	} else {
		tmp = r * ((fma((b * b), -0.16666666666666666, 1.0) * b) / cos((a + b)));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.0082) || !(b <= 2.35e-7))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * b) / cos(Float64(a + b))));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.0082], N[Not[LessEqual[b, 2.35e-7]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00820000000000000069 or 2.35e-7 < b

   1. Initial program 55.6%

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

   3. Taylor expanded in a around 0

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

      1. quot-tan N/A

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

      2. lower-tan.f64 56.4

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

   5. Applied rewrites 56.4%

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

   if -0.00820000000000000069 < b < 2.35e-7

   1. Initial program 98.9%

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

   3. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)}}{\cos \left(a + b\right)} \]
   4. 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 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0082 \lor \neg \left(b \leq 2.35 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-5} \lor \neg \left(b \leq 2.35 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right) \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.6e-5) (not (<= b 2.35e-7)))
   (* r (tan b))
   (/ (* (* (fma (* b b) -0.16666666666666666 1.0) b) r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e-5) || !(b <= 2.35e-7)) {
		tmp = r * tan(b);
	} else {
		tmp = ((fma((b * b), -0.16666666666666666, 1.0) * b) * r) / cos(a);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.6e-5) || !(b <= 2.35e-7))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(Float64(Float64(fma(Float64(b * b), -0.16666666666666666, 1.0) * b) * r) / cos(a));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.6e-5], N[Not[LessEqual[b, 2.35e-7]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * b), $MachinePrecision] * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.59999999999999993e-5 or 2.35e-7 < b

   1. Initial program 55.7%

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

   3. Taylor expanded in a around 0

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

      1. quot-tan N/A

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

      2. lower-tan.f64 56.4

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

   5. Applied rewrites 56.4%

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

   if -1.59999999999999993e-5 < b < 2.35e-7

   1. Initial program 99.4%

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

   3. Taylor expanded in a around inf

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a \cdot \left(1 + \frac{b}{a}\right)\right)}} \]
   4. 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 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 99.4

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

   10. Applied rewrites 99.4%

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

   11. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 99.4

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

   13. Applied rewrites 99.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-5} \lor \neg \left(b \leq 2.35 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(b \cdot b, -0.16666666666666666, 1\right) \cdot b\right) \cdot r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-5} \lor \neg \left(b \leq 2.35 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.6e-5) (not (<= b 2.35e-7)))
   (* r (tan b))
   (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e-5) || !(b <= 2.35e-7)) {
		tmp = r * tan(b);
	} else {
		tmp = (b * r) / cos(a);
	}
	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) :: tmp
    if ((b <= (-1.6d-5)) .or. (.not. (b <= 2.35d-7))) then
        tmp = r * tan(b)
    else
        tmp = (b * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e-5) || !(b <= 2.35e-7)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.6e-5) or not (b <= 2.35e-7):
		tmp = r * math.tan(b)
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.6e-5) || !(b <= 2.35e-7))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(Float64(b * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.6e-5) || ~((b <= 2.35e-7)))
		tmp = r * tan(b);
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.6e-5], N[Not[LessEqual[b, 2.35e-7]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.59999999999999993e-5 or 2.35e-7 < b

   1. Initial program 55.7%

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

   3. Taylor expanded in a around 0

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

      1. quot-tan N/A

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

      2. lower-tan.f64 56.4

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

   5. Applied rewrites 56.4%

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

   if -1.59999999999999993e-5 < b < 2.35e-7

   1. Initial program 99.4%

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

   3. Taylor expanded in a around inf

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(a \cdot \left(1 + \frac{b}{a}\right)\right)}} \]
   4. 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 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 99.4

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

   10. Applied rewrites 99.4%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 99.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-5} \lor \neg \left(b \leq 2.35 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-5} \lor \neg \left(b \leq 2.35 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.6e-5) (not (<= b 2.35e-7)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e-5) || !(b <= 2.35e-7)) {
		tmp = r * tan(b);
	} else {
		tmp = b * (r / cos(a));
	}
	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) :: tmp
    if ((b <= (-1.6d-5)) .or. (.not. (b <= 2.35d-7))) then
        tmp = r * tan(b)
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e-5) || !(b <= 2.35e-7)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.6e-5) or not (b <= 2.35e-7):
		tmp = r * math.tan(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.6e-5) || !(b <= 2.35e-7))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.6e-5) || ~((b <= 2.35e-7)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.6e-5], N[Not[LessEqual[b, 2.35e-7]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.59999999999999993e-5 or 2.35e-7 < b

   1. Initial program 55.7%

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

   3. Taylor expanded in a around 0

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

      1. quot-tan N/A

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

      2. lower-tan.f64 56.4

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

   5. Applied rewrites 56.4%

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

   if -1.59999999999999993e-5 < b < 2.35e-7

   1. Initial program 99.4%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   4. 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 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-5} \lor \neg \left(b \leq 2.35 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ r \cdot \tan b \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r (tan b)))

C

double code(double r, double a, double b) {
	return r * tan(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 * tan(b)
end function

Java

public static double code(double r, double a, double b) {
	return r * Math.tan(b);
}

Python

def code(r, a, b):
	return r * math.tan(b)

Julia

function code(r, a, b)
	return Float64(r * tan(b))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * tan(b);
end

Wolfram

code[r_, a_, b_] := N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in a around 0

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

   1. quot-tan N/A

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

   2. lower-tan.f64 59.4

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

5. Applied rewrites 59.4%

   \[\leadsto r \cdot \color{blue}{\tan b} \]
6. Add Preprocessing

Alternative 14: 34.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, b \cdot b, 0.13333333333333333\right), b \cdot b, 0.3333333333333333\right), b \cdot b, 1\right) \cdot b\right) \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (*
  r
  (*
   (fma
    (fma
     (fma 0.05396825396825397 (* b b) 0.13333333333333333)
     (* b b)
     0.3333333333333333)
    (* b b)
    1.0)
   b)))

C

double code(double r, double a, double b) {
	return r * (fma(fma(fma(0.05396825396825397, (b * b), 0.13333333333333333), (b * b), 0.3333333333333333), (b * b), 1.0) * b);
}

Julia

function code(r, a, b)
	return Float64(r * Float64(fma(fma(fma(0.05396825396825397, Float64(b * b), 0.13333333333333333), Float64(b * b), 0.3333333333333333), Float64(b * b), 1.0) * b))
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[(N[(N[(0.05396825396825397 * N[(b * b), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(b * b), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in a around 0

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

   1. quot-tan N/A

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

   2. lower-tan.f64 59.4

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

5. Applied rewrites 59.4%

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

6. Taylor expanded in b around 0

   \[\leadsto r \cdot \left(b \cdot \color{blue}{\left(1 + {b}^{2} \cdot \left(\frac{1}{3} + {b}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {b}^{2}\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{3} + {b}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {b}^{2}\right)\right)\right) \cdot b\right) \]

   2. lower-*.f64 N/A

      \[\leadsto r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{3} + {b}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {b}^{2}\right)\right)\right) \cdot b\right) \]

   3. +-commutative N/A

      \[\leadsto r \cdot \left(\left({b}^{2} \cdot \left(\frac{1}{3} + {b}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {b}^{2}\right)\right) + 1\right) \cdot b\right) \]

   4. *-commutative N/A

      \[\leadsto r \cdot \left(\left(\left(\frac{1}{3} + {b}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {b}^{2}\right)\right) \cdot {b}^{2} + 1\right) \cdot b\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\frac{1}{3} + {b}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {b}^{2}\right), {b}^{2}, 1\right) \cdot b\right) \]

   6. +-commutative N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left({b}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {b}^{2}\right) + \frac{1}{3}, {b}^{2}, 1\right) \cdot b\right) \]

   7. *-commutative N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\left(\frac{2}{15} + \frac{17}{315} \cdot {b}^{2}\right) \cdot {b}^{2} + \frac{1}{3}, {b}^{2}, 1\right) \cdot b\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15} + \frac{17}{315} \cdot {b}^{2}, {b}^{2}, \frac{1}{3}\right), {b}^{2}, 1\right) \cdot b\right) \]

   9. +-commutative N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{315} \cdot {b}^{2} + \frac{2}{15}, {b}^{2}, \frac{1}{3}\right), {b}^{2}, 1\right) \cdot b\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{315}, {b}^{2}, \frac{2}{15}\right), {b}^{2}, \frac{1}{3}\right), {b}^{2}, 1\right) \cdot b\right) \]

   11. pow2 N/A

       \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{315}, b \cdot b, \frac{2}{15}\right), {b}^{2}, \frac{1}{3}\right), {b}^{2}, 1\right) \cdot b\right) \]

   12. lift-*.f64 N/A

       \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{315}, b \cdot b, \frac{2}{15}\right), {b}^{2}, \frac{1}{3}\right), {b}^{2}, 1\right) \cdot b\right) \]

   13. pow2 N/A

       \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{315}, b \cdot b, \frac{2}{15}\right), b \cdot b, \frac{1}{3}\right), {b}^{2}, 1\right) \cdot b\right) \]

   14. lift-*.f64 N/A

       \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{315}, b \cdot b, \frac{2}{15}\right), b \cdot b, \frac{1}{3}\right), {b}^{2}, 1\right) \cdot b\right) \]

   15. pow2 N/A

       \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{315}, b \cdot b, \frac{2}{15}\right), b \cdot b, \frac{1}{3}\right), b \cdot b, 1\right) \cdot b\right) \]

   16. lift-*.f64 35.7

       \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, b \cdot b, 0.13333333333333333\right), b \cdot b, 0.3333333333333333\right), b \cdot b, 1\right) \cdot b\right) \]

8. Applied rewrites 35.7%

   \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, b \cdot b, 0.13333333333333333\right), b \cdot b, 0.3333333333333333\right), b \cdot b, 1\right) \cdot \color{blue}{b}\right) \]
9. Add Preprocessing

Alternative 15: 34.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, b \cdot b, 0.3333333333333333\right), b \cdot b, 1\right) \cdot b\right) \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (*
  r
  (*
   (fma (fma 0.13333333333333333 (* b b) 0.3333333333333333) (* b b) 1.0)
   b)))

C

double code(double r, double a, double b) {
	return r * (fma(fma(0.13333333333333333, (b * b), 0.3333333333333333), (b * b), 1.0) * b);
}

Julia

function code(r, a, b)
	return Float64(r * Float64(fma(fma(0.13333333333333333, Float64(b * b), 0.3333333333333333), Float64(b * b), 1.0) * b))
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[(N[(0.13333333333333333 * N[(b * b), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in a around 0

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

   1. quot-tan N/A

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

   2. lower-tan.f64 59.4

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

5. Applied rewrites 59.4%

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

6. Taylor expanded in b around 0

   \[\leadsto r \cdot \left(b \cdot \color{blue}{\left(1 + {b}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {b}^{2}\right)\right)}\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {b}^{2}\right)\right) \cdot b\right) \]

   2. lower-*.f64 N/A

      \[\leadsto r \cdot \left(\left(1 + {b}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {b}^{2}\right)\right) \cdot b\right) \]

   3. +-commutative N/A

      \[\leadsto r \cdot \left(\left({b}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {b}^{2}\right) + 1\right) \cdot b\right) \]

   4. *-commutative N/A

      \[\leadsto r \cdot \left(\left(\left(\frac{1}{3} + \frac{2}{15} \cdot {b}^{2}\right) \cdot {b}^{2} + 1\right) \cdot b\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\frac{1}{3} + \frac{2}{15} \cdot {b}^{2}, {b}^{2}, 1\right) \cdot b\right) \]

   6. +-commutative N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\frac{2}{15} \cdot {b}^{2} + \frac{1}{3}, {b}^{2}, 1\right) \cdot b\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, {b}^{2}, \frac{1}{3}\right), {b}^{2}, 1\right) \cdot b\right) \]

   8. pow2 N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, b \cdot b, \frac{1}{3}\right), {b}^{2}, 1\right) \cdot b\right) \]

   9. lift-*.f64 N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, b \cdot b, \frac{1}{3}\right), {b}^{2}, 1\right) \cdot b\right) \]

   10. pow2 N/A

       \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, b \cdot b, \frac{1}{3}\right), b \cdot b, 1\right) \cdot b\right) \]

   11. lift-*.f64 35.6

       \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, b \cdot b, 0.3333333333333333\right), b \cdot b, 1\right) \cdot b\right) \]

8. Applied rewrites 35.6%

   \[\leadsto r \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, b \cdot b, 0.3333333333333333\right), b \cdot b, 1\right) \cdot \color{blue}{b}\right) \]
9. Add Preprocessing

Alternative 16: 34.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ r \cdot \mathsf{fma}\left(b, 1, b \cdot \left(\left(b \cdot b\right) \cdot 0.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (fma b 1.0 (* b (* (* b b) 0.3333333333333333)))))

C

double code(double r, double a, double b) {
	return r * fma(b, 1.0, (b * ((b * b) * 0.3333333333333333)));
}

Julia

function code(r, a, b)
	return Float64(r * fma(b, 1.0, Float64(b * Float64(Float64(b * b) * 0.3333333333333333))))
end

Wolfram

code[r_, a_, b_] := N[(r * N[(b * 1.0 + N[(b * N[(N[(b * b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in a around 0

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

   1. quot-tan N/A

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

   2. lower-tan.f64 59.4

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

5. Applied rewrites 59.4%

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

6. Taylor expanded in b around 0

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

   1. *-commutative N/A

      \[\leadsto r \cdot \left(\left(1 + \frac{1}{3} \cdot {b}^{2}\right) \cdot b\right) \]

   2. lower-*.f64 N/A

      \[\leadsto r \cdot \left(\left(1 + \frac{1}{3} \cdot {b}^{2}\right) \cdot b\right) \]

   3. +-commutative N/A

      \[\leadsto r \cdot \left(\left(\frac{1}{3} \cdot {b}^{2} + 1\right) \cdot b\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\frac{1}{3}, {b}^{2}, 1\right) \cdot b\right) \]

   5. pow2 N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\frac{1}{3}, b \cdot b, 1\right) \cdot b\right) \]

   6. lift-*.f64 35.5

      \[\leadsto r \cdot \left(\mathsf{fma}\left(0.3333333333333333, b \cdot b, 1\right) \cdot b\right) \]

8. Applied rewrites 35.5%

   \[\leadsto r \cdot \left(\mathsf{fma}\left(0.3333333333333333, b \cdot b, 1\right) \cdot \color{blue}{b}\right) \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\frac{1}{3}, b \cdot b, 1\right) \cdot b\right) \]

   2. lift-*.f64 N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\frac{1}{3}, b \cdot b, 1\right) \cdot b\right) \]

   3. lift-fma.f64 N/A

      \[\leadsto r \cdot \left(\left(\frac{1}{3} \cdot \left(b \cdot b\right) + 1\right) \cdot b\right) \]

   4. +-commutative N/A

      \[\leadsto r \cdot \left(\left(1 + \frac{1}{3} \cdot \left(b \cdot b\right)\right) \cdot b\right) \]

   5. pow2 N/A

      \[\leadsto r \cdot \left(\left(1 + \frac{1}{3} \cdot {b}^{2}\right) \cdot b\right) \]

   6. *-commutative N/A

      \[\leadsto r \cdot \left(b \cdot \left(1 + \color{blue}{\frac{1}{3} \cdot {b}^{2}}\right)\right) \]

   7. distribute-lft-in N/A

      \[\leadsto r \cdot \left(b \cdot 1 + b \cdot \color{blue}{\left(\frac{1}{3} \cdot {b}^{2}\right)}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto r \cdot \mathsf{fma}\left(b, 1, b \cdot \left(\frac{1}{3} \cdot {b}^{2}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto r \cdot \mathsf{fma}\left(b, 1, b \cdot \left(\frac{1}{3} \cdot {b}^{2}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto r \cdot \mathsf{fma}\left(b, 1, b \cdot \left({b}^{2} \cdot \frac{1}{3}\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto r \cdot \mathsf{fma}\left(b, 1, b \cdot \left({b}^{2} \cdot \frac{1}{3}\right)\right) \]

   12. pow2 N/A

       \[\leadsto r \cdot \mathsf{fma}\left(b, 1, b \cdot \left(\left(b \cdot b\right) \cdot \frac{1}{3}\right)\right) \]

   13. lift-*.f64 35.5

       \[\leadsto r \cdot \mathsf{fma}\left(b, 1, b \cdot \left(\left(b \cdot b\right) \cdot 0.3333333333333333\right)\right) \]

10. Applied rewrites 35.5%

    \[\leadsto r \cdot \mathsf{fma}\left(b, 1, b \cdot \left(\left(b \cdot b\right) \cdot 0.3333333333333333\right)\right) \]
11. Add Preprocessing

Alternative 17: 34.4% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ r \cdot \left(\mathsf{fma}\left(0.3333333333333333, b \cdot b, 1\right) \cdot b\right) \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (* (fma 0.3333333333333333 (* b b) 1.0) b)))

C

double code(double r, double a, double b) {
	return r * (fma(0.3333333333333333, (b * b), 1.0) * b);
}

Julia

function code(r, a, b)
	return Float64(r * Float64(fma(0.3333333333333333, Float64(b * b), 1.0) * b))
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[(0.3333333333333333 * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in a around 0

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

   1. quot-tan N/A

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

   2. lower-tan.f64 59.4

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

5. Applied rewrites 59.4%

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

6. Taylor expanded in b around 0

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

   1. *-commutative N/A

      \[\leadsto r \cdot \left(\left(1 + \frac{1}{3} \cdot {b}^{2}\right) \cdot b\right) \]

   2. lower-*.f64 N/A

      \[\leadsto r \cdot \left(\left(1 + \frac{1}{3} \cdot {b}^{2}\right) \cdot b\right) \]

   3. +-commutative N/A

      \[\leadsto r \cdot \left(\left(\frac{1}{3} \cdot {b}^{2} + 1\right) \cdot b\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\frac{1}{3}, {b}^{2}, 1\right) \cdot b\right) \]

   5. pow2 N/A

      \[\leadsto r \cdot \left(\mathsf{fma}\left(\frac{1}{3}, b \cdot b, 1\right) \cdot b\right) \]

   6. lift-*.f64 35.5

      \[\leadsto r \cdot \left(\mathsf{fma}\left(0.3333333333333333, b \cdot b, 1\right) \cdot b\right) \]

8. Applied rewrites 35.5%

   \[\leadsto r \cdot \left(\mathsf{fma}\left(0.3333333333333333, b \cdot b, 1\right) \cdot \color{blue}{b}\right) \]
9. Add Preprocessing

Alternative 18: 34.8% accurate, 36.7× 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 78.4%

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

3. Taylor expanded in a around 0

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

   1. quot-tan N/A

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

   2. lower-tan.f64 59.4

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

5. Applied rewrites 59.4%

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

6. Taylor expanded in b around 0

   \[\leadsto r \cdot b \]
7. Step-by-step derivation

8. Applied rewrites 35.5%

   \[\leadsto r \cdot b \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025054 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))