rsin B (should all be same)

Percentage Accurate: 76.6% → 99.5%

Time: 23.1s

Alternatives: 14

Speedup: 1.0×

• Could not converge on a ground truth

  1. r = 2.2911790012119045e+108
  2. a = 7.56289909239326e+262
  3. b = -1.8376674038044385e+224

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

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 74.7%

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

   1. associate-*r/ 74.7%

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

   2. +-commutative 74.7%

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

3. Simplified 74.7%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
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 74.7%

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

   1. associate-*r/ 74.7%

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

   2. +-commutative 74.7%

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

3. Simplified 74.7%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
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 74.7%

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

   1. +-commutative 74.7%

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

3. Simplified 74.7%

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

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

Alternative 4: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-6} \lor \neg \left(b \leq 175\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.1e-6) (not (<= b 175.0)))
   (* r (/ (sin b) (cos b)))
   (* (sin b) (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.1e-6) || !(b <= 175.0)) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = sin(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 <= (-1.1d-6)) .or. (.not. (b <= 175.0d0))) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = sin(b) * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.1e-6) || !(b <= 175.0)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.1e-6) or not (b <= 175.0):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = math.sin(b) * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.1e-6) || !(b <= 175.0))
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.1e-6) || ~((b <= 175.0)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = sin(b) * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.1e-6], N[Not[LessEqual[b, 175.0]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001e-6 or 175 < b

   1. Initial program 53.2%

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

      1. +-commutative 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in a around 0 53.6%

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

      1. associate-/l* 53.6%

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

   7. Simplified 53.6%

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

   if -1.1000000000000001e-6 < b < 175

   1. Initial program 97.7%

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

      1. associate-*r/ 97.7%

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

      2. +-commutative 97.7%

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

   3. Simplified 97.7%

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

      1. expm1-log1p-u 97.7%

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

      2. expm1-undefine 32.3%

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

   6. Applied egg-rr 32.3%

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

      1. expm1-define 97.7%

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

   8. Simplified 97.7%

      \[\leadsto \frac{r \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin b\right)\right)}}{\cos \left(b + a\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 53.9%

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

      2. sqrt-unprod 55.2%

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

      3. pow2 55.2%

         \[\leadsto \frac{r \cdot \mathsf{expm1}\left(\sqrt{\color{blue}{{\left(\mathsf{log1p}\left(\sin b\right)\right)}^{2}}}\right)}{\cos \left(b + a\right)} \]

   10. Applied egg-rr 55.2%

       \[\leadsto \frac{r \cdot \mathsf{expm1}\left(\color{blue}{\sqrt{{\left(\mathsf{log1p}\left(\sin b\right)\right)}^{2}}}\right)}{\cos \left(b + a\right)} \]
   11. Step-by-step derivation

       1. unpow2 55.2%

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

       2. rem-sqrt-square 69.0%

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

   12. Simplified 69.0%

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

   13. Taylor expanded in b around 0 69.0%

       \[\leadsto \frac{r \cdot \mathsf{expm1}\left(\left|\mathsf{log1p}\left(\sin b\right)\right|\right)}{\color{blue}{\cos a}} \]
   14. Step-by-step derivation

       1. *-commutative 69.0%

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

       2. associate-/l* 69.0%

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

       3. add-sqr-sqrt 53.9%

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

       4. fabs-sqr 53.9%

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

       5. add-sqr-sqrt 97.7%

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

       6. expm1-log1p-u 97.7%

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

   15. Applied egg-rr 97.7%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-6} \lor \neg \left(b \leq 175\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-6} \lor \neg \left(b \leq 1.45\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.1e-6) (not (<= b 1.45)))
   (* r (/ (sin b) (cos b)))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.1e-6) || !(b <= 1.45)) {
		tmp = r * (sin(b) / cos(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 <= (-1.1d-6)) .or. (.not. (b <= 1.45d0))) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.1e-6) || !(b <= 1.45)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.1e-6) or not (b <= 1.45):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.1e-6) || !(b <= 1.45))
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.1e-6) || ~((b <= 1.45)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.1e-6], N[Not[LessEqual[b, 1.45]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001e-6 or 1.44999999999999996 < b

   1. Initial program 52.8%

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

      1. +-commutative 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in a around 0 53.2%

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

      1. associate-/l* 53.2%

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

   7. Simplified 53.2%

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

   if -1.1000000000000001e-6 < b < 1.44999999999999996

   1. Initial program 98.4%

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

      1. +-commutative 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in b around 0 98.4%

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

      1. associate-/l* 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-6} \lor \neg \left(b \leq 1.45\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 175:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.1e-6)
   (* (sin b) (/ r (cos b)))
   (if (<= b 175.0) (* (sin b) (/ r (cos a))) (* r (/ (sin b) (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.1e-6) {
		tmp = sin(b) * (r / cos(b));
	} else if (b <= 175.0) {
		tmp = sin(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 <= (-1.1d-6)) then
        tmp = sin(b) * (r / cos(b))
    else if (b <= 175.0d0) then
        tmp = sin(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 <= -1.1e-6) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else if (b <= 175.0) {
		tmp = Math.sin(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 <= -1.1e-6:
		tmp = math.sin(b) * (r / math.cos(b))
	elif b <= 175.0:
		tmp = math.sin(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 <= -1.1e-6)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	elseif (b <= 175.0)
		tmp = Float64(sin(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 <= -1.1e-6)
		tmp = sin(b) * (r / cos(b));
	elseif (b <= 175.0)
		tmp = sin(b) * (r / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.1e-6], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 175.0], N[(N[Sin[b], $MachinePrecision] * 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 < -1.1000000000000001e-6

   1. Initial program 54.8%

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

      1. +-commutative 54.8%

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

   3. Simplified 54.8%

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

      1. associate-*r/ 54.9%

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

      2. clear-num 54.8%

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

   6. Applied egg-rr 54.8%

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

   7. Taylor expanded in a around 0 54.3%

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

      1. clear-num 54.4%

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

      2. *-commutative 54.4%

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

      3. associate-/l* 54.4%

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

   9. Applied egg-rr 54.4%

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

   if -1.1000000000000001e-6 < b < 175

   1. Initial program 97.7%

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

      1. associate-*r/ 97.7%

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

      2. +-commutative 97.7%

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

   3. Simplified 97.7%

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

      1. expm1-log1p-u 97.7%

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

      2. expm1-undefine 32.3%

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

   6. Applied egg-rr 32.3%

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

      1. expm1-define 97.7%

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

   8. Simplified 97.7%

      \[\leadsto \frac{r \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin b\right)\right)}}{\cos \left(b + a\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 53.9%

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

      2. sqrt-unprod 55.2%

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

      3. pow2 55.2%

         \[\leadsto \frac{r \cdot \mathsf{expm1}\left(\sqrt{\color{blue}{{\left(\mathsf{log1p}\left(\sin b\right)\right)}^{2}}}\right)}{\cos \left(b + a\right)} \]

   10. Applied egg-rr 55.2%

       \[\leadsto \frac{r \cdot \mathsf{expm1}\left(\color{blue}{\sqrt{{\left(\mathsf{log1p}\left(\sin b\right)\right)}^{2}}}\right)}{\cos \left(b + a\right)} \]
   11. Step-by-step derivation

       1. unpow2 55.2%

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

       2. rem-sqrt-square 69.0%

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

   12. Simplified 69.0%

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

   13. Taylor expanded in b around 0 69.0%

       \[\leadsto \frac{r \cdot \mathsf{expm1}\left(\left|\mathsf{log1p}\left(\sin b\right)\right|\right)}{\color{blue}{\cos a}} \]
   14. Step-by-step derivation

       1. *-commutative 69.0%

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

       2. associate-/l* 69.0%

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

       3. add-sqr-sqrt 53.9%

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

       4. fabs-sqr 53.9%

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

       5. add-sqr-sqrt 97.7%

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

       6. expm1-log1p-u 97.7%

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

   15. Applied egg-rr 97.7%

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

   if 175 < b

   1. Initial program 51.2%

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

      1. +-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. Taylor expanded in a around 0 52.6%

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

      1. associate-/l* 52.6%

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

   7. Simplified 52.6%

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

Alternative 7: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. associate-*r/ 74.7%

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

   2. +-commutative 74.7%

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

3. Simplified 74.7%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
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. sub-neg 99.5%

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

6. Applied egg-rr 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. fma-define 99.5%

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

   4. *-commutative 99.5%

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

8. Simplified 99.5%

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

   1. associate-/l* 99.5%

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

   2. *-commutative 99.5%

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

   3. fma-undefine 99.4%

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

   4. +-commutative 99.4%

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

   5. *-commutative 99.4%

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

   6. add-sqr-sqrt 48.5%

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

   7. sqrt-unprod 86.9%

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

   8. sqr-neg 86.9%

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

   9. sqrt-unprod 38.4%

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

   10. add-sqr-sqrt 75.0%

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

   11. cos-diff 74.8%

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

10. Applied egg-rr 74.8%

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

11. Final simplification 74.8%

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

Alternative 8: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. associate-*r/ 74.7%

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

   2. +-commutative 74.7%

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

3. Simplified 74.7%

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

   1. *-commutative 74.7%

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

   2. associate-/l* 74.7%

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

6. Applied egg-rr 74.7%

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

Alternative 9: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Final simplification 74.7%

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

Alternative 10: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. +-commutative 74.7%

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

3. Simplified 74.7%

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

5. Taylor expanded in b around 0 53.3%

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

Alternative 11: 55.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.75 \lor \neg \left(b \leq 1.1\right):\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.75) (not (<= b 1.1)))
   (/ 1.0 (/ 1.0 (* r (sin b))))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.75) || !(b <= 1.1)) {
		tmp = 1.0 / (1.0 / (r * sin(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 <= (-0.75d0)) .or. (.not. (b <= 1.1d0))) then
        tmp = 1.0d0 / (1.0d0 / (r * sin(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 <= -0.75) || !(b <= 1.1)) {
		tmp = 1.0 / (1.0 / (r * Math.sin(b)));
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.75) or not (b <= 1.1):
		tmp = 1.0 / (1.0 / (r * math.sin(b)))
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.75) || !(b <= 1.1))
		tmp = Float64(1.0 / Float64(1.0 / Float64(r * sin(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 <= -0.75) || ~((b <= 1.1)))
		tmp = 1.0 / (1.0 / (r * sin(b)));
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.75], N[Not[LessEqual[b, 1.1]], $MachinePrecision]], N[(1.0 / N[(1.0 / N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.75 or 1.1000000000000001 < b

   1. Initial program 52.1%

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

      1. +-commutative 52.1%

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

   3. Simplified 52.1%

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

      1. associate-*r/ 52.1%

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

      2. clear-num 52.1%

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

   6. Applied egg-rr 52.1%

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

   7. Taylor expanded in a around 0 52.4%

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

   8. Taylor expanded in b around 0 12.4%

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

   if -0.75 < b < 1.1000000000000001

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in b around 0 99.1%

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

      1. associate-/l* 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.75 \lor \neg \left(b \leq 1.1\right):\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. +-commutative 74.7%

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

3. Simplified 74.7%

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

5. Taylor expanded in b around 0 49.8%

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

Alternative 13: 51.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. +-commutative 74.7%

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

3. Simplified 74.7%

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

5. Taylor expanded in b around 0 49.6%

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

   1. associate-/l* 49.6%

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

7. Simplified 49.6%

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

Alternative 14: 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 74.7%

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

   1. +-commutative 74.7%

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

3. Simplified 74.7%

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

5. Taylor expanded in b around 0 49.6%

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

   1. associate-/l* 49.6%

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

7. Simplified 49.6%

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

8. Taylor expanded in a around 0 35.5%

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

   1. *-commutative 35.5%

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

10. Simplified 35.5%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))