rsin A (should all be same)

Percentage Accurate: 75.9% → 99.5%

Time: 14.8s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

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

Initial Program: 75.9% accurate, 1.0× speedup

Math

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

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. +-commutative 81.9%

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

3. Simplified 81.9%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*r/ 81.9%

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

   2. *-commutative 81.9%

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

   3. +-commutative 81.9%

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

3. Simplified 81.9%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 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

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) * 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 81.9%

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

   1. associate-*r/ 81.9%

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

   2. *-commutative 81.9%

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

   3. +-commutative 81.9%

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

3. Simplified 81.9%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

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

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

Derivation

1. Initial program 81.9%

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

   1. +-commutative 81.9%

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

3. Simplified 81.9%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 5: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{r \cdot \sin b}{\frac{\cos \left(b + a\right) \cdot -2}{-2} - \sin b \cdot \sin a} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (- (/ (* (cos (+ b a)) -2.0) -2.0) (* (sin b) (sin a)))))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / (((cos((b + a)) * -2.0) / -2.0) - (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((b + a)) * (-2.0d0)) / (-2.0d0)) - (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 + a)) * -2.0) / -2.0) - (Math.sin(b) * Math.sin(a)));
}

Python

def code(r, a, b):
	return (r * math.sin(b)) / (((math.cos((b + a)) * -2.0) / -2.0) - (math.sin(b) * math.sin(a)))

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / Float64(Float64(Float64(cos(Float64(b + a)) * -2.0) / -2.0) - Float64(sin(b) * sin(a))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (r * sin(b)) / (((cos((b + a)) * -2.0) / -2.0) - (sin(b) * sin(a)));
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] / -2.0), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

   1. +-commutative 81.9%

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

3. Simplified 81.9%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.5%

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

   1. cos-mult 83.5%

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

   2. frac-2neg 83.5%

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

   3. cos-diff 82.8%

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

   4. add-sqr-sqrt 42.9%

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

   5. sqrt-unprod 83.0%

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

   6. sqr-neg 83.0%

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

   7. sqrt-unprod 40.1%

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

   8. add-sqr-sqrt 83.7%

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

   9. cancel-sign-sub-inv 83.7%

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

   10. cos-sum 83.5%

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

   11. metadata-eval 83.5%

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

7. Applied egg-rr 83.5%

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

   1. neg-mul-1 83.5%

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

   2. distribute-lft-in 83.5%

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

   3. distribute-rgt-out 83.5%

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

   4. metadata-eval 83.5%

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

9. Simplified 83.5%

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

10. Final simplification 83.5%

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

Alternative 6: 77.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (fma (cos b) (cos a) 0.0))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / fma(cos(b), cos(a), 0.0));
}

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / fma(cos(b), cos(a), 0.0)))
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

   1. associate-*r/ 81.9%

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

   2. *-commutative 81.9%

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

   3. +-commutative 81.9%

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

3. Simplified 81.9%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

   1. add-sqr-sqrt 46.2%

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

   2. sqrt-unprod 88.3%

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

   3. sqr-neg 88.3%

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

   4. sqrt-unprod 42.1%

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

   5. add-sqr-sqrt 81.5%

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

   6. sin-mult 82.6%

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

   7. div-sub 82.6%

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

7. Applied egg-rr 82.6%

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

   1. +-inverses 82.6%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{0}\right)} \cdot r \]

9. Simplified 82.6%

   \[\leadsto \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{0}\right)} \cdot r \]

10. Final simplification 82.6%

    \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \]

Alternative 7: 77.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (fma (cos b) (cos a) 0.0)))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / fma(cos(b), cos(a), 0.0);
}

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / fma(cos(b), cos(a), 0.0))
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

   1. +-commutative 81.9%

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

3. Simplified 81.9%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

   1. add-sqr-sqrt 46.2%

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

   2. sqrt-unprod 88.3%

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

   3. sqr-neg 88.3%

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

   4. sqrt-unprod 42.1%

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

   5. add-sqr-sqrt 81.5%

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

   6. sin-mult 82.6%

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

   7. div-sub 82.6%

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

7. Applied egg-rr 82.6%

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

   1. +-inverses 82.6%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{0}\right)} \cdot r \]

9. Simplified 82.6%

   \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{0}\right)} \]

10. Final simplification 82.6%

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \]

Alternative 8: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 400000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.26e-5)
   (* r (tan b))
   (if (<= b 400000.0) (* b (/ r (cos a))) (* r (/ (sin b) (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.26e-5) {
		tmp = r * tan(b);
	} else if (b <= 400000.0) {
		tmp = b * (r / cos(a));
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	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 <= (-1.26d-5)) then
        tmp = r * tan(b)
    else if (b <= 400000.0d0) then
        tmp = b * (r / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.26e-5) {
		tmp = r * Math.tan(b);
	} else if (b <= 400000.0) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -1.26e-5:
		tmp = r * math.tan(b)
	elif b <= 400000.0:
		tmp = b * (r / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1.26e-5)
		tmp = Float64(r * tan(b));
	elseif (b <= 400000.0)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -1.26e-5)
		tmp = r * tan(b);
	elseif (b <= 400000.0)
		tmp = b * (r / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.26e-5], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 400000.0], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.25999999999999996e-5

   1. Initial program 68.2%

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

      1. associate-*r/ 68.1%

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

      2. *-commutative 68.1%

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

      3. +-commutative 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in a around 0 68.8%

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

      1. expm1-log1p-u 58.0%

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

      2. expm1-udef 57.3%

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

      3. quot-tan 57.4%

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

   6. Applied egg-rr 57.4%

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

      1. expm1-def 58.1%

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

      2. expm1-log1p 68.9%

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

   8. Simplified 68.9%

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

   if -1.25999999999999996e-5 < b < 4e5

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

   3. Simplified 99.0%

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

      1. cos-sum 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. fma-def 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in b around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/l* 98.9%

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

      3. associate-/r/ 99.1%

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

   8. Simplified 99.1%

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

   if 4e5 < b

   1. Initial program 56.9%

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

      1. associate-*r/ 57.0%

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

      2. *-commutative 57.0%

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

      3. +-commutative 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in a around 0 56.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 400000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 9: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 400000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -6.2e-6)
   (/ (sin b) (/ (cos b) r))
   (if (<= b 400000.0) (* b (/ r (cos a))) (* r (/ (sin b) (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -6.2e-6) {
		tmp = sin(b) / (cos(b) / r);
	} else if (b <= 400000.0) {
		tmp = b * (r / cos(a));
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	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 <= (-6.2d-6)) then
        tmp = sin(b) / (cos(b) / r)
    else if (b <= 400000.0d0) then
        tmp = b * (r / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -6.2e-6) {
		tmp = Math.sin(b) / (Math.cos(b) / r);
	} else if (b <= 400000.0) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -6.2e-6:
		tmp = math.sin(b) / (math.cos(b) / r)
	elif b <= 400000.0:
		tmp = b * (r / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -6.2e-6)
		tmp = Float64(sin(b) / Float64(cos(b) / r));
	elseif (b <= 400000.0)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -6.2e-6)
		tmp = sin(b) / (cos(b) / r);
	elseif (b <= 400000.0)
		tmp = b * (r / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -6.2e-6], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 400000.0], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.1999999999999999e-6

   1. Initial program 68.2%

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

      1. +-commutative 68.2%

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

   3. Simplified 68.2%

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

      1. cos-sum 99.3%

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

      2. cancel-sign-sub-inv 99.3%

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

      3. fma-def 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in a around 0 68.9%

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

      1. associate-/l* 68.9%

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

   8. Simplified 68.9%

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

   if -6.1999999999999999e-6 < b < 4e5

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

   3. Simplified 99.0%

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

      1. cos-sum 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. fma-def 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in b around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/l* 98.9%

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

      3. associate-/r/ 99.1%

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

   8. Simplified 99.1%

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

   if 4e5 < b

   1. Initial program 56.9%

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

      1. associate-*r/ 57.0%

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

      2. *-commutative 57.0%

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

      3. +-commutative 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in a around 0 56.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 400000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 10: 75.9% 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 81.9%

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

   1. associate-/l* 81.7%

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

   2. +-commutative 81.7%

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

3. Simplified 81.7%

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

   1. associate-/r/ 81.9%

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

5. Applied egg-rr 81.9%

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

6. Final simplification 81.9%

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

Alternative 11: 76.0% 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 81.9%

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

   1. associate-*r/ 81.9%

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

   2. *-commutative 81.9%

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

   3. +-commutative 81.9%

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

3. Simplified 81.9%

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

4. Final simplification 81.9%

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

Alternative 12: 75.9% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 81.9%

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

2. Final simplification 81.9%

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

Alternative 13: 75.9% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 81.9%

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

   1. +-commutative 81.9%

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

3. Simplified 81.9%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

   1. add-sqr-sqrt 46.2%

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

   2. sqrt-unprod 88.4%

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

   3. sqr-neg 88.4%

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

   4. sqrt-unprod 42.1%

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

   5. add-sqr-sqrt 81.5%

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

   6. fma-def 81.5%

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

   7. cos-diff 82.0%

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

7. Applied egg-rr 82.0%

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

8. Final simplification 82.0%

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

Alternative 14: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{-5} \lor \neg \left(b \leq 400000\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.26e-5) (not (<= b 400000.0)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.26e-5) || !(b <= 400000.0)) {
		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 <= (-1.26d-5)) .or. (.not. (b <= 400000.0d0))) 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 <= -1.26e-5) || !(b <= 400000.0)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.26e-5) or not (b <= 400000.0):
		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 <= -1.26e-5) || !(b <= 400000.0))
		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 <= -1.26e-5) || ~((b <= 400000.0)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.26e-5], N[Not[LessEqual[b, 400000.0]], $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 < -1.25999999999999996e-5 or 4e5 < b

   1. Initial program 62.2%

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

      1. associate-*r/ 62.2%

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

      2. *-commutative 62.2%

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

      3. +-commutative 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in a around 0 62.4%

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

      1. expm1-log1p-u 51.1%

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

      2. expm1-udef 50.7%

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

      3. quot-tan 50.7%

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

   6. Applied egg-rr 50.7%

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

      1. expm1-def 51.1%

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

      2. expm1-log1p 62.4%

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

   8. Simplified 62.4%

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

   if -1.25999999999999996e-5 < b < 4e5

   1. Initial program 99.0%

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

      1. associate-*r/ 99.0%

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

      2. *-commutative 99.0%

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

      3. +-commutative 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in b around 0 99.1%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{-5} \lor \neg \left(b \leq 400000\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 15: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-6} \lor \neg \left(b \leq 400000\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -3.6e-6) (not (<= b 400000.0)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -3.6e-6) || !(b <= 400000.0)) {
		tmp = r * tan(b);
	} else {
		tmp = b * (r / 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 <= (-3.6d-6)) .or. (.not. (b <= 400000.0d0))) 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 <= -3.6e-6) || !(b <= 400000.0)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -3.6e-6) or not (b <= 400000.0):
		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 <= -3.6e-6) || !(b <= 400000.0))
		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 <= -3.6e-6) || ~((b <= 400000.0)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -3.6e-6], N[Not[LessEqual[b, 400000.0]], $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 < -3.59999999999999984e-6 or 4e5 < b

   1. Initial program 62.2%

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

      1. associate-*r/ 62.2%

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

      2. *-commutative 62.2%

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

      3. +-commutative 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in a around 0 62.4%

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

      1. expm1-log1p-u 51.1%

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

      2. expm1-udef 50.7%

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

      3. quot-tan 50.7%

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

   6. Applied egg-rr 50.7%

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

      1. expm1-def 51.1%

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

      2. expm1-log1p 62.4%

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

   8. Simplified 62.4%

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

   if -3.59999999999999984e-6 < b < 4e5

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

   3. Simplified 99.0%

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

      1. cos-sum 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. fma-def 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in b around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/l* 98.9%

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

      3. associate-/r/ 99.1%

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

   8. Simplified 99.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-6} \lor \neg \left(b \leq 400000\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 16: 38.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. +-commutative 81.9%

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

3. Simplified 81.9%

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

4. Taylor expanded in b around 0 56.4%

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

   1. mul-1-neg 56.4%

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

   2. unsub-neg 56.4%

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

6. Simplified 56.4%

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

7. Taylor expanded in a around 0 44.8%

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

8. Final simplification 44.8%

   \[\leadsto r \cdot \sin b \]

Alternative 17: 59.9% 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 81.9%

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

   1. associate-*r/ 81.9%

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

   2. *-commutative 81.9%

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

   3. +-commutative 81.9%

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

3. Simplified 81.9%

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

4. Taylor expanded in a around 0 67.3%

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

   1. expm1-log1p-u 62.1%

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

   2. expm1-udef 36.9%

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

   3. quot-tan 36.9%

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

6. Applied egg-rr 36.9%

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

   1. expm1-def 62.1%

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

   2. expm1-log1p 67.3%

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

8. Simplified 67.3%

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

9. Final simplification 67.3%

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

Alternative 18: 34.7% 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 81.9%

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

   1. associate-*r/ 81.9%

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

   2. *-commutative 81.9%

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

   3. +-commutative 81.9%

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

3. Simplified 81.9%

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

4. Taylor expanded in b around 0 55.2%

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

5. Taylor expanded in a around 0 40.6%

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

6. Final simplification 40.6%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023238 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))