rsin B (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 18.6s

Alternatives: 9

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.8% 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

real(8) function code(r, a, b)
    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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. associate-*r/ 74.9%

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

   2. /-rgt-identity 74.9%

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

   3. metadata-eval 74.9%

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

   4. associate-/l/ 74.9%

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

   5. associate-*l/ 74.9%

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

   6. associate-/l/ 74.9%

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

   7. metadata-eval 74.9%

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

   8. metadata-eval 74.9%

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

   9. metadata-eval 74.9%

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

   10. /-rgt-identity 74.9%

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

   11. +-commutative 74.9%

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

3. Simplified 74.9%

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

   1. cos-sum 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. prod-diff 99.4%

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

5. Applied egg-rr 99.4%

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

   1. fma-udef 99.4%

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

   2. distribute-lft-neg-in 99.4%

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

   3. cancel-sign-sub-inv 99.4%

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

   4. *-commutative 99.4%

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

   5. *-rgt-identity 99.4%

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

   6. *-commutative 99.4%

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

   7. fma-udef 99.4%

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

   8. *-rgt-identity 99.4%

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

   9. distribute-lft-neg-in 99.4%

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

   10. *-rgt-identity 99.4%

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

   11. fma-udef 99.5%

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

   12. *-commutative 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in r around 0 99.5%

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

   1. fma-def 99.5%

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

   2. *-commutative 99.5%

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

10. Applied egg-rr 99.5%

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

    1. fma-udef 99.5%

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

    2. neg-mul-1 99.5%

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

    3. *-commutative 99.5%

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

    4. distribute-lft-neg-in 99.5%

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

    5. fma-udef 99.5%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. +-commutative 74.9%

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

3. Simplified 74.9%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. associate-*r/ 74.9%

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

   2. /-rgt-identity 74.9%

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

   3. metadata-eval 74.9%

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

   4. associate-/l/ 74.9%

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

   5. associate-*l/ 74.9%

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

   6. associate-/l/ 74.9%

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

   7. metadata-eval 74.9%

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

   8. metadata-eval 74.9%

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

   9. metadata-eval 74.9%

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

   10. /-rgt-identity 74.9%

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

   11. +-commutative 74.9%

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

3. Simplified 74.9%

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

   1. cos-sum 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. prod-diff 99.4%

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

5. Applied egg-rr 99.4%

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

   1. fma-udef 99.4%

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

   2. distribute-lft-neg-in 99.4%

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

   3. cancel-sign-sub-inv 99.4%

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

   4. *-commutative 99.4%

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

   5. *-rgt-identity 99.4%

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

   6. *-commutative 99.4%

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

   7. fma-udef 99.4%

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

   8. *-rgt-identity 99.4%

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

   9. distribute-lft-neg-in 99.4%

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

   10. *-rgt-identity 99.4%

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

   11. fma-udef 99.5%

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

   12. *-commutative 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in r around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. mul-1-neg 99.5%

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

   3. sub-neg 99.5%

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

   4. *-commutative 99.5%

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

10. Applied egg-rr 99.5%

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

11. Final simplification 99.5%

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

Alternative 4: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

2. Final simplification 74.9%

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

Alternative 5: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. associate-*r/ 74.9%

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

   2. /-rgt-identity 74.9%

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

   3. metadata-eval 74.9%

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

   4. associate-/l/ 74.9%

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

   5. associate-*l/ 74.9%

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

   6. associate-/l/ 74.9%

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

   7. metadata-eval 74.9%

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

   8. metadata-eval 74.9%

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

   9. metadata-eval 74.9%

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

   10. /-rgt-identity 74.9%

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

   11. +-commutative 74.9%

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

3. Simplified 74.9%

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

4. Final simplification 74.9%

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

Alternative 6: 76.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-6} \lor \neg \left(b \leq 1.2 \cdot 10^{-12}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -4.5e-6) (not (<= b 1.2e-12)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.5e-6) || !(b <= 1.2e-12)) {
		tmp = r * tan(b);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-4.5d-6)) .or. (.not. (b <= 1.2d-12))) then
        tmp = r * tan(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.5e-6) || !(b <= 1.2e-12)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -4.5e-6) or not (b <= 1.2e-12):
		tmp = r * math.tan(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -4.5e-6) || !(b <= 1.2e-12))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -4.5e-6) || ~((b <= 1.2e-12)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -4.5e-6], N[Not[LessEqual[b, 1.2e-12]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.50000000000000011e-6 or 1.19999999999999994e-12 < b

   1. Initial program 50.0%

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

      1. +-commutative 50.0%

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

   3. Simplified 50.0%

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

      1. log1p-expm1-u 49.0%

         \[\leadsto r \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin b}{\cos \left(b + a\right)}\right)\right)} \]

   5. Applied egg-rr 49.0%

      \[\leadsto r \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin b}{\cos \left(b + a\right)}\right)\right)} \]

   6. Taylor expanded in a around 0 48.4%

      \[\leadsto r \cdot \mathsf{log1p}\left(\color{blue}{e^{\frac{\sin b}{\cos b}} - 1}\right) \]
   7. Step-by-step derivation

      1. expm1-def 49.1%

         \[\leadsto r \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\sin b}{\cos b}\right)}\right) \]

      2. log1p-expm1-u 50.2%

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

      3. add-log-exp 49.4%

         \[\leadsto r \cdot \color{blue}{\log \left(e^{\frac{\sin b}{\cos b}}\right)} \]

      4. *-un-lft-identity 49.4%

         \[\leadsto r \cdot \log \color{blue}{\left(1 \cdot e^{\frac{\sin b}{\cos b}}\right)} \]

      5. log-prod 49.4%

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

      6. metadata-eval 49.4%

         \[\leadsto r \cdot \left(\color{blue}{0} + \log \left(e^{\frac{\sin b}{\cos b}}\right)\right) \]

      7. add-log-exp 50.2%

         \[\leadsto r \cdot \left(0 + \color{blue}{\frac{\sin b}{\cos b}}\right) \]

      8. quot-tan 50.3%

         \[\leadsto r \cdot \left(0 + \color{blue}{\tan b}\right) \]

   8. Applied egg-rr 50.3%

      \[\leadsto r \cdot \color{blue}{\left(0 + \tan b\right)} \]
   9. Step-by-step derivation

      1. +-lft-identity 50.3%

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

   10. Simplified 50.3%

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

   if -4.50000000000000011e-6 < b < 1.19999999999999994e-12

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 99.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-6} \lor \neg \left(b \leq 1.2 \cdot 10^{-12}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 7: 33.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ r \cdot \mathsf{log1p}\left(b\right) \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r (log1p b)))

C

double code(double r, double a, double b) {
	return r * log1p(b);
}

Java

public static double code(double r, double a, double b) {
	return r * Math.log1p(b);
}

Python

def code(r, a, b):
	return r * math.log1p(b)

Julia

function code(r, a, b)
	return Float64(r * log1p(b))
end

Wolfram

code[r_, a_, b_] := N[(r * N[Log[1 + b], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.9%

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

   1. +-commutative 74.9%

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

3. Simplified 74.9%

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

   1. log1p-expm1-u 74.4%

      \[\leadsto r \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin b}{\cos \left(b + a\right)}\right)\right)} \]

5. Applied egg-rr 74.4%

   \[\leadsto r \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin b}{\cos \left(b + a\right)}\right)\right)} \]

6. Taylor expanded in a around 0 36.8%

   \[\leadsto r \cdot \mathsf{log1p}\left(\color{blue}{e^{\frac{\sin b}{\cos b}} - 1}\right) \]

7. Taylor expanded in b around 0 35.0%

   \[\leadsto r \cdot \mathsf{log1p}\left(\color{blue}{b}\right) \]

8. Final simplification 35.0%

   \[\leadsto r \cdot \mathsf{log1p}\left(b\right) \]

Alternative 8: 60.5% accurate, 2.0× 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

real(8) function code(r, a, b)
    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 74.9%

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

   1. +-commutative 74.9%

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

3. Simplified 74.9%

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

   1. log1p-expm1-u 74.4%

      \[\leadsto r \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin b}{\cos \left(b + a\right)}\right)\right)} \]

5. Applied egg-rr 74.4%

   \[\leadsto r \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin b}{\cos \left(b + a\right)}\right)\right)} \]

6. Taylor expanded in a around 0 36.8%

   \[\leadsto r \cdot \mathsf{log1p}\left(\color{blue}{e^{\frac{\sin b}{\cos b}} - 1}\right) \]
7. Step-by-step derivation

   1. expm1-def 56.9%

      \[\leadsto r \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\sin b}{\cos b}\right)}\right) \]

   2. log1p-expm1-u 57.4%

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

   3. add-log-exp 37.3%

      \[\leadsto r \cdot \color{blue}{\log \left(e^{\frac{\sin b}{\cos b}}\right)} \]

   4. *-un-lft-identity 37.3%

      \[\leadsto r \cdot \log \color{blue}{\left(1 \cdot e^{\frac{\sin b}{\cos b}}\right)} \]

   5. log-prod 37.3%

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

   6. metadata-eval 37.3%

      \[\leadsto r \cdot \left(\color{blue}{0} + \log \left(e^{\frac{\sin b}{\cos b}}\right)\right) \]

   7. add-log-exp 57.4%

      \[\leadsto r \cdot \left(0 + \color{blue}{\frac{\sin b}{\cos b}}\right) \]

   8. quot-tan 57.5%

      \[\leadsto r \cdot \left(0 + \color{blue}{\tan b}\right) \]

8. Applied egg-rr 57.5%

   \[\leadsto r \cdot \color{blue}{\left(0 + \tan b\right)} \]
9. Step-by-step derivation

   1. +-lft-identity 57.5%

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

10. Simplified 57.5%

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

11. Final simplification 57.5%

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

Alternative 9: 33.9% accurate, 69.0× 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

real(8) function code(r, a, b)
    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 74.9%

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

   1. +-commutative 74.9%

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

3. Simplified 74.9%

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

4. Taylor expanded in b around 0 52.4%

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

5. Taylor expanded in a around 0 34.8%

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

6. Final simplification 34.8%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023333 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))