rsin A (should all be same)

Percentage Accurate: 77.0% → 99.5%

Time: 14.8s

Alternatives: 15

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: 77.0% 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

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

FPCore

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

C

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

Julia

function code(r, a, b)
	return Float64(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 75.7%

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

   1. +-commutative 75.7%

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

3. Simplified 75.7%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
4. Add Preprocessing
5. 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-define 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. Applied egg-rr 99.5%

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

Alternative 2: 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 75.7%

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

   1. +-commutative 75.7%

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

3. Simplified 75.7%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

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 75.7%

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

   1. associate-/l* 75.6%

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

   2. remove-double-neg 75.6%

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

   3. remove-double-neg 75.6%

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

   4. +-commutative 75.6%

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

3. Simplified 75.6%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 4: 78.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. +-commutative 75.7%

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

3. Simplified 75.7%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

   1. sin-mult 77.2%

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

   2. div-sub 77.2%

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

   3. cos-diff 76.9%

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

   4. add-sqr-sqrt 41.2%

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

   5. sqrt-unprod 76.1%

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

   6. sqr-neg 76.1%

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

   7. sqrt-unprod 35.0%

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

   8. add-sqr-sqrt 75.1%

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

   9. cancel-sign-sub-inv 75.1%

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

   10. cos-sum 77.0%

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

8. Applied egg-rr 77.0%

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

   1. +-inverses 77.0%

      \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{0}} \]

10. Simplified 77.0%

    \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{0}} \]

11. Final simplification 77.0%

    \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a} \]
12. Add Preprocessing

Alternative 5: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-6}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 220:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -2.05e-6)
   (/ r (/ (cos b) (sin b)))
   (if (<= b 220.0) (/ (* r (sin b)) (cos a)) (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.05e-6) {
		tmp = r / (cos(b) / sin(b));
	} else if (b <= 220.0) {
		tmp = (r * sin(b)) / cos(a);
	} else {
		tmp = r * tan(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 <= (-2.05d-6)) then
        tmp = r / (cos(b) / sin(b))
    else if (b <= 220.0d0) then
        tmp = (r * sin(b)) / cos(a)
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.05e-6) {
		tmp = r / (Math.cos(b) / Math.sin(b));
	} else if (b <= 220.0) {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -2.05e-6:
		tmp = r / (math.cos(b) / math.sin(b))
	elif b <= 220.0:
		tmp = (r * math.sin(b)) / math.cos(a)
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -2.05e-6)
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	elseif (b <= 220.0)
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -2.05e-6)
		tmp = r / (cos(b) / sin(b));
	elseif (b <= 220.0)
		tmp = (r * sin(b)) / cos(a);
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -2.05e-6], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 220.0], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0499999999999999e-6

   1. Initial program 53.5%

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

      1. associate-/l* 53.4%

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

      2. remove-double-neg 53.4%

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

      3. remove-double-neg 53.4%

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

      4. +-commutative 53.4%

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

   3. Simplified 53.4%

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

      1. clear-num 53.3%

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

      2. un-div-inv 53.5%

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

   6. Applied egg-rr 53.5%

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

   7. Taylor expanded in a around 0 52.8%

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

   if -2.0499999999999999e-6 < b < 220

   1. Initial program 97.8%

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

      1. +-commutative 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in b around 0 97.8%

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

   if 220 < b

   1. Initial program 51.7%

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

      1. associate-/l* 51.7%

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

      2. remove-double-neg 51.7%

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

      3. remove-double-neg 51.7%

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

      4. +-commutative 51.7%

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

   3. Simplified 51.7%

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

      1. log1p-expm1-u 51.6%

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

   6. Applied egg-rr 51.6%

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

   7. Taylor expanded in a around 0 52.8%

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

      1. expm1-define 53.0%

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

      2. log1p-expm1-u 53.0%

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

      3. add-log-exp 52.9%

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

      4. *-un-lft-identity 52.9%

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

      5. log-prod 52.9%

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

      6. metadata-eval 52.9%

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

      7. add-log-exp 53.0%

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

      8. quot-tan 53.2%

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

   9. Applied egg-rr 53.2%

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

       1. +-lft-identity 53.2%

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

   11. Simplified 53.2%

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

Alternative 6: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.86 \cdot 10^{-6}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 0.00017:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.86e-6)
   (/ r (/ (cos b) (sin b)))
   (if (<= b 0.00017) (/ (* r b) (cos a)) (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.86e-6) {
		tmp = r / (cos(b) / sin(b));
	} else if (b <= 0.00017) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = r * tan(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 <= (-1.86d-6)) then
        tmp = r / (cos(b) / sin(b))
    else if (b <= 0.00017d0) then
        tmp = (r * b) / cos(a)
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.86e-6) {
		tmp = r / (Math.cos(b) / Math.sin(b));
	} else if (b <= 0.00017) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -1.86e-6:
		tmp = r / (math.cos(b) / math.sin(b))
	elif b <= 0.00017:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1.86e-6)
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	elseif (b <= 0.00017)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -1.86e-6)
		tmp = r / (cos(b) / sin(b));
	elseif (b <= 0.00017)
		tmp = (r * b) / cos(a);
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.86e-6], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.00017], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.86e-6

   1. Initial program 53.5%

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

      1. associate-/l* 53.4%

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

      2. remove-double-neg 53.4%

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

      3. remove-double-neg 53.4%

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

      4. +-commutative 53.4%

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

   3. Simplified 53.4%

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

      1. clear-num 53.3%

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

      2. un-div-inv 53.5%

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

   6. Applied egg-rr 53.5%

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

   7. Taylor expanded in a around 0 52.8%

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

   if -1.86e-6 < b < 1.7e-4

   1. Initial program 98.4%

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

      1. associate-/l* 98.3%

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

      2. remove-double-neg 98.3%

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

      3. remove-double-neg 98.3%

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

      4. +-commutative 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in b around 0 98.4%

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

   if 1.7e-4 < b

   1. Initial program 51.2%

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

      1. associate-/l* 51.2%

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

      2. remove-double-neg 51.2%

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

      3. remove-double-neg 51.2%

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

      4. +-commutative 51.2%

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

   3. Simplified 51.2%

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

      1. log1p-expm1-u 51.2%

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

   6. Applied egg-rr 51.2%

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

   7. Taylor expanded in a around 0 52.3%

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

      1. expm1-define 52.5%

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

      2. log1p-expm1-u 52.5%

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

      3. add-log-exp 52.4%

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

      4. *-un-lft-identity 52.4%

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

      5. log-prod 52.4%

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

      6. metadata-eval 52.4%

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

      7. add-log-exp 52.5%

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

      8. quot-tan 52.8%

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

   9. Applied egg-rr 52.8%

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

       1. +-lft-identity 52.8%

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

   11. Simplified 52.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.86 \cdot 10^{-6}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 0.00017:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% 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 75.7%

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

3. Final simplification 75.7%

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

Alternative 8: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double r, double a, double b) {
	return r / (cos((b + a)) / 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 / (cos((b + a)) / sin(b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. associate-/l* 75.6%

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

   2. remove-double-neg 75.6%

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

   3. remove-double-neg 75.6%

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

   4. +-commutative 75.6%

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

3. Simplified 75.6%

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

   1. clear-num 75.5%

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

   2. un-div-inv 75.6%

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

6. Applied egg-rr 75.6%

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

Alternative 9: 77.0% 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 75.7%

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

   1. +-commutative 75.7%

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

3. Simplified 75.7%

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

   1. *-commutative 75.7%

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

   2. associate-/l* 75.6%

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

6. Applied egg-rr 75.6%

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

Alternative 10: 77.0% 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 75.7%

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

   1. associate-/l* 75.6%

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

   2. remove-double-neg 75.6%

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

   3. remove-double-neg 75.6%

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

   4. +-commutative 75.6%

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

3. Simplified 75.6%

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

Alternative 11: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-6} \lor \neg \left(b \leq 0.00031\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -2.4e-6) (not (<= b 0.00031)))
   (* r (tan b))
   (/ (* r b) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -2.4e-6) || !(b <= 0.00031)) {
		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 <= (-2.4d-6)) .or. (.not. (b <= 0.00031d0))) 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 <= -2.4e-6) || !(b <= 0.00031)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = (r * b) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -2.4e-6) or not (b <= 0.00031):
		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 <= -2.4e-6) || !(b <= 0.00031))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(Float64(r * b) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -2.4e-6) || ~((b <= 0.00031)))
		tmp = r * tan(b);
	else
		tmp = (r * b) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -2.4e-6], N[Not[LessEqual[b, 0.00031]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.3999999999999999e-6 or 3.1e-4 < b

   1. Initial program 52.2%

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

      1. associate-/l* 52.2%

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

      2. remove-double-neg 52.2%

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

      3. remove-double-neg 52.2%

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

      4. +-commutative 52.2%

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

   3. Simplified 52.2%

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

      1. log1p-expm1-u 51.5%

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

   6. Applied egg-rr 51.5%

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

   7. Taylor expanded in a around 0 51.6%

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

      1. expm1-define 51.9%

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

      2. log1p-expm1-u 52.6%

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

      3. add-log-exp 52.3%

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

      4. *-un-lft-identity 52.3%

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

      5. log-prod 52.3%

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

      6. metadata-eval 52.3%

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

      7. add-log-exp 52.6%

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

      8. quot-tan 52.8%

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

   9. Applied egg-rr 52.8%

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

       1. +-lft-identity 52.8%

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

   11. Simplified 52.8%

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

   if -2.3999999999999999e-6 < b < 3.1e-4

   1. Initial program 98.4%

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

      1. associate-/l* 98.3%

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

      2. remove-double-neg 98.3%

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

      3. remove-double-neg 98.3%

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

      4. +-commutative 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in b around 0 98.4%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-6} \lor \neg \left(b \leq 0.00031\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-6} \lor \neg \left(b \leq 0.000205\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -2.3e-6) (not (<= b 0.000205)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -2.3e-6) || !(b <= 0.000205)) {
		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 <= (-2.3d-6)) .or. (.not. (b <= 0.000205d0))) 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 <= -2.3e-6) || !(b <= 0.000205)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -2.3e-6) or not (b <= 0.000205):
		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 <= -2.3e-6) || !(b <= 0.000205))
		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 <= -2.3e-6) || ~((b <= 0.000205)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -2.3e-6], N[Not[LessEqual[b, 0.000205]], $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 < -2.3e-6 or 2.05e-4 < b

   1. Initial program 52.2%

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

      1. associate-/l* 52.2%

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

      2. remove-double-neg 52.2%

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

      3. remove-double-neg 52.2%

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

      4. +-commutative 52.2%

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

   3. Simplified 52.2%

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

      1. log1p-expm1-u 51.5%

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

   6. Applied egg-rr 51.5%

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

   7. Taylor expanded in a around 0 51.6%

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

      1. expm1-define 51.9%

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

      2. log1p-expm1-u 52.6%

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

      3. add-log-exp 52.3%

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

      4. *-un-lft-identity 52.3%

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

      5. log-prod 52.3%

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

      6. metadata-eval 52.3%

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

      7. add-log-exp 52.6%

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

      8. quot-tan 52.8%

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

   9. Applied egg-rr 52.8%

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

       1. +-lft-identity 52.8%

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

   11. Simplified 52.8%

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

   if -2.3e-6 < b < 2.05e-4

   1. Initial program 98.4%

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

      1. +-commutative 98.4%

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

   3. Simplified 98.4%

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

      1. cos-sum 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in b around 0 98.4%

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

      1. associate-/l* 98.3%

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

   9. Simplified 98.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-6} \lor \neg \left(b \leq 0.000205\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.2% 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 75.7%

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

   1. associate-/l* 75.6%

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

   2. remove-double-neg 75.6%

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

   3. remove-double-neg 75.6%

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

   4. +-commutative 75.6%

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

3. Simplified 75.6%

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

   1. log1p-expm1-u 75.2%

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

6. Applied egg-rr 75.2%

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

7. Taylor expanded in a around 0 38.6%

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

   1. expm1-define 58.0%

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

   2. log1p-expm1-u 58.3%

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

   3. add-log-exp 39.0%

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

   4. *-un-lft-identity 39.0%

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

   5. log-prod 39.0%

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

   6. metadata-eval 39.0%

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

   7. add-log-exp 58.3%

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

   8. quot-tan 58.4%

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

9. Applied egg-rr 58.4%

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

    1. +-lft-identity 58.4%

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

11. Simplified 58.4%

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

Alternative 14: 34.4% 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 75.7%

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

   1. associate-/l* 75.6%

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

   2. remove-double-neg 75.6%

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

   3. remove-double-neg 75.6%

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

   4. +-commutative 75.6%

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

3. Simplified 75.6%

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

   1. log1p-expm1-u 75.2%

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

6. Applied egg-rr 75.2%

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

7. Taylor expanded in a around 0 38.6%

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

8. Taylor expanded in b around 0 35.0%

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

Alternative 15: 34.4% 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 75.7%

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

   1. associate-/l* 75.6%

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

   2. remove-double-neg 75.6%

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

   3. remove-double-neg 75.6%

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

   4. +-commutative 75.6%

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

3. Simplified 75.6%

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

5. Taylor expanded in b around 0 52.2%

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

6. Taylor expanded in a around 0 34.9%

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

Reproduce

herbie shell --seed 2024095 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))