rsin B (should all be same)

Percentage Accurate: 76.1% → 99.5%

Time: 17.2s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. remove-double-neg 75.9%

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

   2. remove-double-neg 75.9%

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

   3. +-commutative 75.9%

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

3. Simplified 75.9%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 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 75.9%

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

   1. remove-double-neg 75.9%

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

   2. remove-double-neg 75.9%

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

   3. +-commutative 75.9%

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

3. Simplified 75.9%

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

   1. cos-sum 99.5%

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

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. Final simplification 99.5%

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

Alternative 3: 77.1% accurate, 0.7× speedup

Math

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

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

Derivation

1. Initial program 75.9%

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

   1. remove-double-neg 75.9%

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

   2. remove-double-neg 75.9%

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

   3. +-commutative 75.9%

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

3. Simplified 75.9%

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

   1. add-log-exp 75.7%

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

6. Applied egg-rr 75.7%

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

   1. add-sqr-sqrt 75.5%

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

   2. sqrt-unprod 75.7%

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

   3. prod-exp 75.7%

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

   4. +-commutative 75.7%

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

   5. cos-sum 76.9%

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

   6. *-commutative 76.9%

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

   7. cancel-sign-sub-inv 76.9%

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

   8. *-commutative 76.9%

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

   9. add-sqr-sqrt 37.4%

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

   10. sqrt-unprod 76.1%

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

   11. sqr-neg 76.1%

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

   12. sqrt-unprod 38.8%

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

   13. add-sqr-sqrt 75.4%

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

   14. cos-diff 75.6%

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

8. Applied egg-rr 77.5%

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

9. Final simplification 77.5%

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

Alternative 4: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-7} \lor \neg \left(a \leq 1.55 \cdot 10^{-30}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -4.9e-7) (not (<= a 1.55e-30)))
   (* r (/ (sin b) (cos a)))
   (* r (tan b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -4.9e-7) || !(a <= 1.55e-30)) {
		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 ((a <= (-4.9d-7)) .or. (.not. (a <= 1.55d-30))) 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 ((a <= -4.9e-7) || !(a <= 1.55e-30)) {
		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 (a <= -4.9e-7) or not (a <= 1.55e-30):
		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 ((a <= -4.9e-7) || !(a <= 1.55e-30))
		tmp = Float64(r * Float64(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 ((a <= -4.9e-7) || ~((a <= 1.55e-30)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -4.9e-7], N[Not[LessEqual[a, 1.55e-30]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.8999999999999997e-7 or 1.54999999999999995e-30 < a

   1. Initial program 53.4%

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

      1. remove-double-neg 53.4%

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

      2. remove-double-neg 53.4%

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

      3. +-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. Taylor expanded in b around 0 53.7%

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

   if -4.8999999999999997e-7 < a < 1.54999999999999995e-30

   1. Initial program 99.6%

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

      1. remove-double-neg 99.6%

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

      2. remove-double-neg 99.6%

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

      3. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. add-log-exp 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in a around 0 99.3%

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

      1. expm1-log1p-u 91.4%

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

      2. expm1-udef 52.5%

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

      3. rem-log-exp 52.5%

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

      4. quot-tan 52.5%

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

   9. Applied egg-rr 52.5%

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

   11. Simplified 99.7%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-7} \lor \neg \left(a \leq 1.55 \cdot 10^{-30}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-30}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -4.9e-7)
   (/ (* r (sin b)) (cos a))
   (if (<= a 1.55e-30) (* r (tan b)) (* r (/ (sin b) (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -4.9e-7) {
		tmp = (r * sin(b)) / cos(a);
	} else if (a <= 1.55e-30) {
		tmp = r * tan(b);
	} else {
		tmp = r * (sin(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 (a <= (-4.9d-7)) then
        tmp = (r * sin(b)) / cos(a)
    else if (a <= 1.55d-30) then
        tmp = r * tan(b)
    else
        tmp = r * (sin(b) / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -4.9e-7) {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	} else if (a <= 1.55e-30) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (Math.sin(b) / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -4.9e-7:
		tmp = (r * math.sin(b)) / math.cos(a)
	elif a <= 1.55e-30:
		tmp = r * math.tan(b)
	else:
		tmp = r * (math.sin(b) / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -4.9e-7)
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	elseif (a <= 1.55e-30)
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -4.9e-7)
		tmp = (r * sin(b)) / cos(a);
	elseif (a <= 1.55e-30)
		tmp = r * tan(b);
	else
		tmp = r * (sin(b) / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -4.9e-7], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.55e-30], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.8999999999999997e-7

   1. Initial program 50.6%

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

      1. associate-*r/ 50.6%

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

      2. +-commutative 50.6%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in b around 0 50.8%

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

   if -4.8999999999999997e-7 < a < 1.54999999999999995e-30

   1. Initial program 99.6%

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

      1. remove-double-neg 99.6%

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

      2. remove-double-neg 99.6%

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

      3. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. add-log-exp 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in a around 0 99.3%

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

      1. expm1-log1p-u 91.4%

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

      2. expm1-udef 52.5%

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

      3. rem-log-exp 52.5%

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

      4. quot-tan 52.5%

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

   9. Applied egg-rr 52.5%

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

   11. Simplified 99.7%

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

   if 1.54999999999999995e-30 < a

   1. Initial program 56.2%

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

      1. remove-double-neg 56.2%

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

      2. remove-double-neg 56.2%

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

      3. +-commutative 56.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in b around 0 56.6%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-30}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% 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 75.9%

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

3. Final simplification 75.9%

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

Alternative 7: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-5} \lor \neg \left(b \leq 1.6 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -3.1e-5) (not (<= b 1.6e-5)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -3.1e-5) || !(b <= 1.6e-5)) {
		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 <= (-3.1d-5)) .or. (.not. (b <= 1.6d-5))) 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 <= -3.1e-5) || !(b <= 1.6e-5)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -3.1e-5) or not (b <= 1.6e-5):
		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 <= -3.1e-5) || !(b <= 1.6e-5))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -3.1e-5) || ~((b <= 1.6e-5)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -3.1e-5], N[Not[LessEqual[b, 1.6e-5]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.10000000000000014e-5 or 1.59999999999999993e-5 < b

   1. Initial program 51.8%

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

      1. remove-double-neg 51.8%

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

      2. remove-double-neg 51.8%

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

      3. +-commutative 51.8%

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

   3. Simplified 51.8%

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

      1. add-log-exp 51.5%

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

   6. Applied egg-rr 51.5%

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

   7. Taylor expanded in a around 0 50.9%

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

      1. expm1-log1p-u 42.7%

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

      2. expm1-udef 42.4%

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

      3. rem-log-exp 42.4%

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

      4. quot-tan 42.4%

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

   9. Applied egg-rr 42.4%

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

   11. Simplified 51.3%

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

   if -3.10000000000000014e-5 < b < 1.59999999999999993e-5

   1. Initial program 99.0%

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

      1. remove-double-neg 99.0%

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

      2. remove-double-neg 99.0%

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

      3. +-commutative 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in b around 0 99.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-5} \lor \neg \left(b \leq 1.6 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.4% 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 75.9%

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

   1. remove-double-neg 75.9%

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

   2. remove-double-neg 75.9%

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

   3. +-commutative 75.9%

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

3. Simplified 75.9%

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

   1. add-log-exp 75.7%

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

6. Applied egg-rr 75.7%

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

7. Taylor expanded in a around 0 60.1%

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

   1. expm1-log1p-u 56.0%

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

   2. expm1-udef 34.0%

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

   3. rem-log-exp 34.0%

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

   4. quot-tan 34.0%

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

9. Applied egg-rr 34.0%

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

11. Simplified 60.2%

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

12. Final simplification 60.2%

    \[\leadsto r \cdot \tan b \]
13. Add Preprocessing

Alternative 9: 34.6% 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 75.9%

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

   1. remove-double-neg 75.9%

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

   2. remove-double-neg 75.9%

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

   3. +-commutative 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in b around 0 52.9%

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

6. Taylor expanded in a around 0 37.2%

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

7. Final simplification 37.2%

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

Reproduce

herbie shell --seed 2024017 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))