rsin B (should all be same)

Percentage Accurate: 76.4% → 99.5%

Time: 25.1s

Alternatives: 14

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: 76.4% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-neg-in N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. sin-neg N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. neg-lowering-neg.f64 N/A

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

   10. sin-lowering-sin.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. cos-lowering-cos.f64 N/A

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

   13. cos-lowering-cos.f64 99.6

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in r around 0

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

   1. /-lowering-/.f64 N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. sin-lowering-sin.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. sin-lowering-sin.f64 N/A

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

   10. neg-mul-1 N/A

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

   11. sin-lowering-sin.f64 N/A

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

   12. neg-mul-1 N/A

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

   13. neg-lowering-neg.f64 99.6

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

7. Simplified 99.6%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. sin-lowering-sin.f64 N/A

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

   5. neg-lowering-neg.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. cos-lowering-cos.f64 N/A

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

   9. cos-lowering-cos.f64 99.6

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

9. Applied egg-rr 99.6%

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

Alternative 2: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos \left(b + a\right)}\\ t_1 := r \cdot \tan b\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (sin b) (cos (+ b a)))) (t_1 (* r (tan b))))
   (if (<= t_0 -2e-7) t_1 (if (<= t_0 0.05) (* b (/ r (cos a))) t_1))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) / cos((b + a));
	double t_1 = r * tan(b);
	double tmp;
	if (t_0 <= -2e-7) {
		tmp = t_1;
	} else if (t_0 <= 0.05) {
		tmp = b * (r / cos(a));
	} else {
		tmp = t_1;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(b) / cos((b + a))
    t_1 = r * tan(b)
    if (t_0 <= (-2d-7)) then
        tmp = t_1
    else if (t_0 <= 0.05d0) then
        tmp = b * (r / cos(a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) / Math.cos((b + a));
	double t_1 = r * Math.tan(b);
	double tmp;
	if (t_0 <= -2e-7) {
		tmp = t_1;
	} else if (t_0 <= 0.05) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) / math.cos((b + a))
	t_1 = r * math.tan(b)
	tmp = 0
	if t_0 <= -2e-7:
		tmp = t_1
	elif t_0 <= 0.05:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_1
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) / cos(Float64(b + a)))
	t_1 = Float64(r * tan(b))
	tmp = 0.0
	if (t_0 <= -2e-7)
		tmp = t_1;
	elseif (t_0 <= 0.05)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) / cos((b + a));
	t_1 = r * tan(b);
	tmp = 0.0;
	if (t_0 <= -2e-7)
		tmp = t_1;
	elseif (t_0 <= 0.05)
		tmp = b * (r / cos(a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-7], t$95$1, If[LessEqual[t$95$0, 0.05], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -1.9999999999999999e-7 or 0.050000000000000003 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 57.6%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 58.4

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

   5. Simplified 58.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 58.4

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

   7. Applied egg-rr 58.4%

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

   if -1.9999999999999999e-7 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < 0.050000000000000003

   1. Initial program 99.3%

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

   3. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. cos-lowering-cos.f64 99.2

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

   5. Simplified 99.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(b + a\right)} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;\frac{\sin b}{\cos \left(b + a\right)} \leq 0.05:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-neg-in N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. sin-neg N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. neg-lowering-neg.f64 N/A

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

   10. sin-lowering-sin.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. cos-lowering-cos.f64 N/A

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

   13. cos-lowering-cos.f64 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / fma(cos(b), cos(a), Float64(sin(a) * Float64(-sin(b))))))
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[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.4%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. neg-lowering-neg.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. sin-lowering-sin.f64 N/A

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

   11. sin-lowering-sin.f64 99.6

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -260000:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 2050:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -260000.0)
   (* r (tan b))
   (if (<= b 2050.0)
     (*
      r
      (/
       (fma
        (fma
         b
         (* b (fma (* b b) -0.0001984126984126984 0.008333333333333333))
         -0.16666666666666666)
        (* b (* b b))
        b)
       (cos (+ b a))))
     (/ (* (sin b) r) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -260000.0) {
		tmp = r * tan(b);
	} else if (b <= 2050.0) {
		tmp = r * (fma(fma(b, (b * fma((b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (b * (b * b)), b) / cos((b + a)));
	} else {
		tmp = (sin(b) * r) / cos(b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -260000.0)
		tmp = Float64(r * tan(b));
	elseif (b <= 2050.0)
		tmp = Float64(r * Float64(fma(fma(b, Float64(b * fma(Float64(b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(b * Float64(b * b)), b) / cos(Float64(b + a))));
	else
		tmp = Float64(Float64(sin(b) * r) / cos(b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -260000.0], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2050.0], N[(r * N[(N[(N[(b * N[(b * N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.6e5

   1. Initial program 53.8%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 55.7

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

   5. Simplified 55.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 55.9

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

   7. Applied egg-rr 55.9%

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

   if -2.6e5 < b < 2050

   1. Initial program 98.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(a + b\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto r \cdot \frac{b \cdot \color{blue}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) + 1\right)}}{\cos \left(a + b\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) + b \cdot 1}}{\cos \left(a + b\right)} \]

      3. associate-*r* N/A

         \[\leadsto r \cdot \frac{\color{blue}{\left(b \cdot {b}^{2}\right) \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)} + b \cdot 1}{\cos \left(a + b\right)} \]

      4. *-commutative N/A

         \[\leadsto r \cdot \frac{\color{blue}{\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot \left(b \cdot {b}^{2}\right)} + b \cdot 1}{\cos \left(a + b\right)} \]

      5. *-rgt-identity N/A

         \[\leadsto r \cdot \frac{\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot \left(b \cdot {b}^{2}\right) + \color{blue}{b}}{\cos \left(a + b\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, b \cdot {b}^{2}, b\right)}}{\cos \left(a + b\right)} \]

   5. Simplified 98.0%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}}{\cos \left(a + b\right)} \]

   if 2050 < b

   1. Initial program 61.9%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 62.0

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

   5. Simplified 62.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -260000:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 2050:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 79.4%

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. sin-lowering-sin.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. +-commutative N/A

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

   7. +-lowering-+.f64 79.4

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

4. Applied egg-rr 79.4%

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

5. Final simplification 79.4%

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

Alternative 7: 76.4% 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 79.4%

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

3. Final simplification 79.4%

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

Alternative 8: 75.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -260000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2050:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -260000.0)
     t_0
     (if (<= b 2050.0)
       (*
        r
        (/
         (fma
          (fma
           b
           (* b (fma (* b b) -0.0001984126984126984 0.008333333333333333))
           -0.16666666666666666)
          (* b (* b b))
          b)
         (cos (+ b a))))
       (/ 1.0 (/ 1.0 t_0))))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -260000.0) {
		tmp = t_0;
	} else if (b <= 2050.0) {
		tmp = r * (fma(fma(b, (b * fma((b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (b * (b * b)), b) / cos((b + a)));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -260000.0)
		tmp = t_0;
	elseif (b <= 2050.0)
		tmp = Float64(r * Float64(fma(fma(b, Float64(b * fma(Float64(b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(b * Float64(b * b)), b) / cos(Float64(b + a))));
	else
		tmp = Float64(1.0 / Float64(1.0 / t_0));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -260000.0], t$95$0, If[LessEqual[b, 2050.0], N[(r * N[(N[(N[(b * N[(b * N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.6e5

   1. Initial program 53.8%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 55.7

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

   5. Simplified 55.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 55.9

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

   7. Applied egg-rr 55.9%

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

   if -2.6e5 < b < 2050

   1. Initial program 98.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(a + b\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto r \cdot \frac{b \cdot \color{blue}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) + 1\right)}}{\cos \left(a + b\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) + b \cdot 1}}{\cos \left(a + b\right)} \]

      3. associate-*r* N/A

         \[\leadsto r \cdot \frac{\color{blue}{\left(b \cdot {b}^{2}\right) \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)} + b \cdot 1}{\cos \left(a + b\right)} \]

      4. *-commutative N/A

         \[\leadsto r \cdot \frac{\color{blue}{\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot \left(b \cdot {b}^{2}\right)} + b \cdot 1}{\cos \left(a + b\right)} \]

      5. *-rgt-identity N/A

         \[\leadsto r \cdot \frac{\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot \left(b \cdot {b}^{2}\right) + \color{blue}{b}}{\cos \left(a + b\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, b \cdot {b}^{2}, b\right)}}{\cos \left(a + b\right)} \]

   5. Simplified 98.0%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}}{\cos \left(a + b\right)} \]

   if 2050 < b

   1. Initial program 61.9%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 62.0

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

   5. Simplified 62.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. quot-tan N/A

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

      8. tan-lowering-tan.f64 61.8

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

   7. Applied egg-rr 61.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{r \cdot \tan b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -260000:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 2050:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -260000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.04:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -260000.0)
     t_0
     (if (<= b 0.04)
       (* r (/ (fma b (* b (* b -0.16666666666666666)) b) (cos (+ b a))))
       (/ 1.0 (/ 1.0 t_0))))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -260000.0) {
		tmp = t_0;
	} else if (b <= 0.04) {
		tmp = r * (fma(b, (b * (b * -0.16666666666666666)), b) / cos((b + a)));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -260000.0)
		tmp = t_0;
	elseif (b <= 0.04)
		tmp = Float64(r * Float64(fma(b, Float64(b * Float64(b * -0.16666666666666666)), b) / cos(Float64(b + a))));
	else
		tmp = Float64(1.0 / Float64(1.0 / t_0));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -260000.0], t$95$0, If[LessEqual[b, 0.04], N[(r * N[(N[(b * N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.6e5

   1. Initial program 53.8%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 55.7

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

   5. Simplified 55.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 55.9

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

   7. Applied egg-rr 55.9%

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

   if -2.6e5 < b < 0.0400000000000000008

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)}}{\cos \left(a + b\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto r \cdot \frac{b \cdot \color{blue}{\left(\frac{-1}{6} \cdot {b}^{2} + 1\right)}}{\cos \left(a + b\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + b \cdot 1}}{\cos \left(a + b\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto r \cdot \frac{b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + \color{blue}{b}}{\cos \left(a + b\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(b, \frac{-1}{6} \cdot {b}^{2}, b\right)}}{\cos \left(a + b\right)} \]

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 98.6

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(b, b \cdot \color{blue}{\left(b \cdot -0.16666666666666666\right)}, b\right)}{\cos \left(a + b\right)} \]

   5. Simplified 98.6%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}}{\cos \left(a + b\right)} \]

   if 0.0400000000000000008 < b

   1. Initial program 61.5%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 61.6

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

   5. Simplified 61.6%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. quot-tan N/A

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

      8. tan-lowering-tan.f64 61.4

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

   7. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{r \cdot \tan b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -260000:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 0.04:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.04:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -3.2e-7)
     t_0
     (if (<= b 0.04) (* r (/ b (cos a))) (/ 1.0 (/ 1.0 t_0))))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -3.2e-7) {
		tmp = t_0;
	} else if (b <= 0.04) {
		tmp = r * (b / cos(a));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = r * tan(b)
    if (b <= (-3.2d-7)) then
        tmp = t_0
    else if (b <= 0.04d0) then
        tmp = r * (b / cos(a))
    else
        tmp = 1.0d0 / (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.tan(b);
	double tmp;
	if (b <= -3.2e-7) {
		tmp = t_0;
	} else if (b <= 0.04) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -3.2e-7:
		tmp = t_0
	elif b <= 0.04:
		tmp = r * (b / math.cos(a))
	else:
		tmp = 1.0 / (1.0 / t_0)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -3.2e-7)
		tmp = t_0;
	elseif (b <= 0.04)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(1.0 / Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -3.2e-7)
		tmp = t_0;
	elseif (b <= 0.04)
		tmp = r * (b / cos(a));
	else
		tmp = 1.0 / (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.2e-7], t$95$0, If[LessEqual[b, 0.04], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.2000000000000001e-7

   1. Initial program 54.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 55.6

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

   5. Simplified 55.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 55.7

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

   7. Applied egg-rr 55.7%

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

   if -3.2000000000000001e-7 < b < 0.0400000000000000008

   1. Initial program 99.3%

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

   3. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 N/A

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

      2. cos-lowering-cos.f64 99.3

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

   5. Simplified 99.3%

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

   if 0.0400000000000000008 < b

   1. Initial program 61.5%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 61.6

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

   5. Simplified 61.6%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. quot-tan N/A

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

      8. tan-lowering-tan.f64 61.4

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

   7. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{r \cdot \tan b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 0.04:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.04:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -3.2e-7) t_0 (if (<= b 0.04) (* r (/ b (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -3.2e-7) {
		tmp = t_0;
	} else if (b <= 0.04) {
		tmp = r * (b / cos(a));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = r * tan(b)
    if (b <= (-3.2d-7)) then
        tmp = t_0
    else if (b <= 0.04d0) then
        tmp = r * (b / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.tan(b);
	double tmp;
	if (b <= -3.2e-7) {
		tmp = t_0;
	} else if (b <= 0.04) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -3.2e-7:
		tmp = t_0
	elif b <= 0.04:
		tmp = r * (b / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -3.2e-7)
		tmp = t_0;
	elseif (b <= 0.04)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -3.2e-7)
		tmp = t_0;
	elseif (b <= 0.04)
		tmp = r * (b / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.2e-7], t$95$0, If[LessEqual[b, 0.04], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.2000000000000001e-7 or 0.0400000000000000008 < b

   1. Initial program 57.6%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 58.4

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

   5. Simplified 58.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 58.4

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

   7. Applied egg-rr 58.4%

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

   if -3.2000000000000001e-7 < b < 0.0400000000000000008

   1. Initial program 99.3%

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

   3. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 N/A

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

      2. cos-lowering-cos.f64 99.3

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

   5. Simplified 99.3%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 0.04:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.8% accurate, 2.1× 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 79.4%

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

3. Taylor expanded in a around 0

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

   1. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. sin-lowering-sin.f64 N/A

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

   4. cos-lowering-cos.f64 61.4

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

5. Simplified 61.4%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. quot-tan N/A

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

   5. tan-lowering-tan.f64 61.4

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

7. Applied egg-rr 61.4%

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

8. Final simplification 61.4%

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

Alternative 13: 38.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Taylor expanded in b around 0

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

   1. cos-lowering-cos.f64 57.4

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

5. Simplified 57.4%

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

6. Taylor expanded in a around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sin-lowering-sin.f64 39.5

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

8. Simplified 39.5%

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

9. Final simplification 39.5%

   \[\leadsto \sin b \cdot r \]
10. Add Preprocessing

Alternative 14: 34.4% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

def code(r, a, b):
	return b * r

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Taylor expanded in b around 0

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. cos-lowering-cos.f64 54.0

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

5. Simplified 54.0%

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

6. Taylor expanded in a around 0

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

   1. *-lowering-*.f64 35.6

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

8. Simplified 35.6%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))