VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 7.6s

Alternatives: 9

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	return -(x * (1.0 / tan(B))) + (1.0 / sin(B));
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + (1.0d0 / sin(b))
end function

Java

public static double code(double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
}

Python

def code(B, x):
	return -(x * (1.0 / math.tan(B))) + (1.0 / math.sin(B))

Julia

function code(B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / sin(B)))
end

MATLAB

function tmp = code(B, x)
	tmp = -(x * (1.0 / tan(B))) + (1.0 / sin(B));
end

Wolfram

code[B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	return -(x * (1.0 / tan(B))) + (1.0 / sin(B));
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + (1.0d0 / sin(b))
end function

Java

public static double code(double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
}

Python

def code(B, x):
	return -(x * (1.0 / math.tan(B))) + (1.0 / math.sin(B))

Julia

function code(B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / sin(B)))
end

MATLAB

function tmp = code(B, x)
	tmp = -(x * (1.0 / tan(B))) + (1.0 / sin(B));
end

Wolfram

code[B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{\tan B} + {\sin B}^{-1} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (+ (/ (- x) (tan B)) (pow (sin B) -1.0)))

C

double code(double B, double x) {
	return (-x / tan(B)) + pow(sin(B), -1.0);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-x / tan(b)) + (sin(b) ** (-1.0d0))
end function

Java

public static double code(double B, double x) {
	return (-x / Math.tan(B)) + Math.pow(Math.sin(B), -1.0);
}

Python

def code(B, x):
	return (-x / math.tan(B)) + math.pow(math.sin(B), -1.0)

Julia

function code(B, x)
	return Float64(Float64(Float64(-x) / tan(B)) + (sin(B) ^ -1.0))
end

MATLAB

function tmp = code(B, x)
	tmp = (-x / tan(B)) + (sin(B) ^ -1.0);
end

Wolfram

code[B_, x_] := N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{1}{\sin B} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{1}{\sin B} \]

   3. associate-*r/ N/A

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{\sin B} \]

   4. *-rgt-identity N/A

      \[\leadsto \left(-\frac{\color{blue}{x}}{\tan B}\right) + \frac{1}{\sin B} \]

   5. lower-/.f64 99.8

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{\sin B} \]

4. Applied rewrites 99.8%

   \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{\sin B} \]

5. Final simplification 99.8%

   \[\leadsto \frac{-x}{\tan B} + {\sin B}^{-1} \]
6. Add Preprocessing

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -24000000000000 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -24000000000000.0) (not (<= x 1.15)))
   (+ (/ (- x) (tan B)) (pow B -1.0))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -24000000000000.0) || !(x <= 1.15)) {
		tmp = (-x / tan(B)) + pow(B, -1.0);
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-24000000000000.0d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = (-x / tan(b)) + (b ** (-1.0d0))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -24000000000000.0) || !(x <= 1.15)) {
		tmp = (-x / Math.tan(B)) + Math.pow(B, -1.0);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -24000000000000.0) or not (x <= 1.15):
		tmp = (-x / math.tan(B)) + math.pow(B, -1.0)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -24000000000000.0) || !(x <= 1.15))
		tmp = Float64(Float64(Float64(-x) / tan(B)) + (B ^ -1.0));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -24000000000000.0) || ~((x <= 1.15)))
		tmp = (-x / tan(B)) + (B ^ -1.0);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -24000000000000.0], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e13 or 1.1499999999999999 < x

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

      2. inv-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\sin B}^{-1}} \]

      3. sqr-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\sin B}^{\left(\frac{-1}{2}\right)} \cdot {\sin B}^{\left(\frac{-1}{2}\right)}} \]

      4. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left({\sin B}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left({\sin B}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\color{blue}{\left({\sin B}^{\left(\frac{-1}{2}\right)}\right)}}^{2} \]

      7. metadata-eval 52.3

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left({\sin B}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 52.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left({\sin B}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 99.5

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]

   7. Applied rewrites 99.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{1}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{1}{B} \]

      3. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{B} \]

      4. *-rgt-identity N/A

         \[\leadsto \left(-\frac{\color{blue}{x}}{\tan B}\right) + \frac{1}{B} \]

      5. lower-/.f64 99.7

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{B} \]

   9. Applied rewrites 99.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{B} \]

   if -2.4e13 < x < 1.1499999999999999

   1. Initial program 99.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{1}{\sin B} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{1}{\sin B} \]

      3. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{\sin B} \]

      4. *-rgt-identity N/A

         \[\leadsto \left(-\frac{\color{blue}{x}}{\tan B}\right) + \frac{1}{\sin B} \]

      5. lower-/.f64 99.8

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{\sin B} \]

   4. Applied rewrites 99.8%

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{\sin B} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{\tan B}\right) + \frac{1}{\sin B}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\tan B}\right)\right)} + \frac{1}{\sin B} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

      4. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\tan B}} + \frac{1}{\sin B} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\tan B} + \color{blue}{\frac{1}{\sin B}} \]

      6. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\tan B}\right)\right)} + \frac{1}{\sin B} \]

      7. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\color{blue}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

      8. tan-quot N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\color{blue}{\frac{\sin B}{\cos B}}}\right)\right) + \frac{1}{\sin B} \]

      9. lift-sin.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\frac{\color{blue}{\sin B}}{\cos B}}\right)\right) + \frac{1}{\sin B} \]

      10. lift-cos.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) + \frac{1}{\sin B} \]

      11. associate-/r/ N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\sin B} \cdot \cos B}\right)\right) + \frac{1}{\sin B} \]

      12. associate-*l/ N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \cos B}{\sin B}}\right)\right) + \frac{1}{\sin B} \]

      13. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\cos B \cdot x}}{\sin B}\right)\right) + \frac{1}{\sin B} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\cos B \cdot x}}{\sin B}\right)\right) + \frac{1}{\sin B} \]

      15. distribute-neg-frac N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\cos B \cdot x\right)}{\sin B}} + \frac{1}{\sin B} \]

      16. div-add-rev N/A

          \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\cos B \cdot x\right)\right) + 1}{\sin B}} \]

      17. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\cos B \cdot x\right)\right) + 1}{\sin B}} \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-x, \cos B, 1\right)}{\sin B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{\color{blue}{1 + -1 \cdot x}}{\sin B} \]
   8. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\color{blue}{1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot x}}{\sin B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1 - \color{blue}{1} \cdot x}{\sin B} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{1 - \color{blue}{x}}{\sin B} \]

      4. lower--.f64 97.7

         \[\leadsto \frac{\color{blue}{1 - x}}{\sin B} \]

   9. Applied rewrites 97.7%

      \[\leadsto \frac{\color{blue}{1 - x}}{\sin B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -24000000000000 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos B \cdot x}{\sin B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (- 1.0 (* (cos B) x)) (sin B)))

C

double code(double B, double x) {
	return (1.0 - (cos(B) * x)) / sin(B);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 - (cos(b) * x)) / sin(b)
end function

Java

public static double code(double B, double x) {
	return (1.0 - (Math.cos(B) * x)) / Math.sin(B);
}

Python

def code(B, x):
	return (1.0 - (math.cos(B) * x)) / math.sin(B)

Julia

function code(B, x)
	return Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B))
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 - (cos(B) * x)) / sin(B);
end

Wolfram

code[B_, x_] := N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
4. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \frac{1}{\sin B} - \color{blue}{1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   2. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{x \cdot \cos B}{\sin B}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{1}{\sin B} + \color{blue}{-1} \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{1}{\sin B} + \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

   5. div-add-rev N/A

      \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

   7. associate-*r* N/A

      \[\leadsto \frac{1 + \color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

   8. mul-1-neg N/A

      \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \cos B}{\sin B} \]

   9. fp-cancel-sub-sign N/A

      \[\leadsto \frac{\color{blue}{1 - x \cdot \cos B}}{\sin B} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\color{blue}{1 - x \cdot \cos B}}{\sin B} \]

   11. *-commutative N/A

       \[\leadsto \frac{1 - \color{blue}{\cos B \cdot x}}{\sin B} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1 - \color{blue}{\cos B \cdot x}}{\sin B} \]

   13. lower-cos.f64 N/A

       \[\leadsto \frac{1 - \color{blue}{\cos B} \cdot x}{\sin B} \]

   14. lower-sin.f64 99.7

       \[\leadsto \frac{1 - \cos B \cdot x}{\color{blue}{\sin B}} \]

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]
6. Add Preprocessing

Alternative 4: 51.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(-\frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right) \cdot x}{B}\right) + {B}^{-1} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (- (/ (* (fma -0.3333333333333333 (* B B) 1.0) x) B)) (pow B -1.0)))

C

double code(double B, double x) {
	return -((fma(-0.3333333333333333, (B * B), 1.0) * x) / B) + pow(B, -1.0);
}

Julia

function code(B, x)
	return Float64(Float64(-Float64(Float64(fma(-0.3333333333333333, Float64(B * B), 1.0) * x) / B)) + (B ^ -1.0))
end

Wolfram

code[B_, x_] := N[((-N[(N[(N[(-0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / B), $MachinePrecision]) + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   2. inv-pow N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\sin B}^{-1}} \]

   3. sqr-pow N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\sin B}^{\left(\frac{-1}{2}\right)} \cdot {\sin B}^{\left(\frac{-1}{2}\right)}} \]

   4. pow2 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left({\sin B}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left({\sin B}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\color{blue}{\left({\sin B}^{\left(\frac{-1}{2}\right)}\right)}}^{2} \]

   7. metadata-eval 53.2

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left({\sin B}^{\color{blue}{-0.5}}\right)}^{2} \]

4. Applied rewrites 53.2%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left({\sin B}^{-0.5}\right)}^{2}} \]

5. Taylor expanded in B around 0

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
6. Step-by-step derivation

   1. lower-/.f64 74.1

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]

7. Applied rewrites 74.1%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]

8. Taylor expanded in B around 0

   \[\leadsto \left(-\color{blue}{\frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B}}\right) + \frac{1}{B} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(-\frac{x + \color{blue}{\left({B}^{2} \cdot x\right) \cdot \frac{-1}{3}}}{B}\right) + \frac{1}{B} \]

   2. associate-*r* N/A

      \[\leadsto \left(-\frac{x + \color{blue}{{B}^{2} \cdot \left(x \cdot \frac{-1}{3}\right)}}{B}\right) + \frac{1}{B} \]

   3. metadata-eval N/A

      \[\leadsto \left(-\frac{x + {B}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{2} - \frac{-1}{6}\right)}\right)}{B}\right) + \frac{1}{B} \]

   4. distribute-rgt-out-- N/A

      \[\leadsto \left(-\frac{x + {B}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}}{B}\right) + \frac{1}{B} \]

   5. lower-/.f64 N/A

      \[\leadsto \left(-\color{blue}{\frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{B}}\right) + \frac{1}{B} \]

   6. distribute-rgt-out-- N/A

      \[\leadsto \left(-\frac{x + {B}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{2} - \frac{-1}{6}\right)\right)}}{B}\right) + \frac{1}{B} \]

   7. metadata-eval N/A

      \[\leadsto \left(-\frac{x + {B}^{2} \cdot \left(x \cdot \color{blue}{\frac{-1}{3}}\right)}{B}\right) + \frac{1}{B} \]

   8. associate-*r* N/A

      \[\leadsto \left(-\frac{x + \color{blue}{\left({B}^{2} \cdot x\right) \cdot \frac{-1}{3}}}{B}\right) + \frac{1}{B} \]

   9. *-commutative N/A

      \[\leadsto \left(-\frac{x + \color{blue}{\frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}}{B}\right) + \frac{1}{B} \]

   10. associate-*r* N/A

       \[\leadsto \left(-\frac{x + \color{blue}{\left(\frac{-1}{3} \cdot {B}^{2}\right) \cdot x}}{B}\right) + \frac{1}{B} \]

   11. distribute-rgt1-in N/A

       \[\leadsto \left(-\frac{\color{blue}{\left(\frac{-1}{3} \cdot {B}^{2} + 1\right) \cdot x}}{B}\right) + \frac{1}{B} \]

   12. +-commutative N/A

       \[\leadsto \left(-\frac{\color{blue}{\left(1 + \frac{-1}{3} \cdot {B}^{2}\right)} \cdot x}{B}\right) + \frac{1}{B} \]

   13. lower-*.f64 N/A

       \[\leadsto \left(-\frac{\color{blue}{\left(1 + \frac{-1}{3} \cdot {B}^{2}\right) \cdot x}}{B}\right) + \frac{1}{B} \]

   14. +-commutative N/A

       \[\leadsto \left(-\frac{\color{blue}{\left(\frac{-1}{3} \cdot {B}^{2} + 1\right)} \cdot x}{B}\right) + \frac{1}{B} \]

   15. lower-fma.f64 N/A

       \[\leadsto \left(-\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {B}^{2}, 1\right)} \cdot x}{B}\right) + \frac{1}{B} \]

   16. unpow2 N/A

       \[\leadsto \left(-\frac{\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{B \cdot B}, 1\right) \cdot x}{B}\right) + \frac{1}{B} \]

   17. lower-*.f64 52.0

       \[\leadsto \left(-\frac{\mathsf{fma}\left(-0.3333333333333333, \color{blue}{B \cdot B}, 1\right) \cdot x}{B}\right) + \frac{1}{B} \]

10. Applied rewrites 52.0%

    \[\leadsto \left(-\color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right) \cdot x}{B}}\right) + \frac{1}{B} \]

11. Final simplification 52.0%

    \[\leadsto \left(-\frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right) \cdot x}{B}\right) + {B}^{-1} \]
12. Add Preprocessing

Alternative 5: 76.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{\sin B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (- 1.0 x) (sin B)))

C

double code(double B, double x) {
	return (1.0 - x) / sin(B);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 - x) / sin(b)
end function

Java

public static double code(double B, double x) {
	return (1.0 - x) / Math.sin(B);
}

Python

def code(B, x):
	return (1.0 - x) / math.sin(B)

Julia

function code(B, x)
	return Float64(Float64(1.0 - x) / sin(B))
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 - x) / sin(B);
end

Wolfram

code[B_, x_] := N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{1}{\sin B} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{1}{\sin B} \]

   3. associate-*r/ N/A

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{\sin B} \]

   4. *-rgt-identity N/A

      \[\leadsto \left(-\frac{\color{blue}{x}}{\tan B}\right) + \frac{1}{\sin B} \]

   5. lower-/.f64 99.8

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{\sin B} \]

4. Applied rewrites 99.8%

   \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{\sin B} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(-\frac{x}{\tan B}\right) + \frac{1}{\sin B}} \]

   2. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\tan B}\right)\right)} + \frac{1}{\sin B} \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

   4. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\tan B}} + \frac{1}{\sin B} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\tan B} + \color{blue}{\frac{1}{\sin B}} \]

   6. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\tan B}\right)\right)} + \frac{1}{\sin B} \]

   7. lift-tan.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\color{blue}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

   8. tan-quot N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\color{blue}{\frac{\sin B}{\cos B}}}\right)\right) + \frac{1}{\sin B} \]

   9. lift-sin.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\frac{\color{blue}{\sin B}}{\cos B}}\right)\right) + \frac{1}{\sin B} \]

   10. lift-cos.f64 N/A

       \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) + \frac{1}{\sin B} \]

   11. associate-/r/ N/A

       \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\sin B} \cdot \cos B}\right)\right) + \frac{1}{\sin B} \]

   12. associate-*l/ N/A

       \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \cos B}{\sin B}}\right)\right) + \frac{1}{\sin B} \]

   13. *-commutative N/A

       \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\cos B \cdot x}}{\sin B}\right)\right) + \frac{1}{\sin B} \]

   14. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\cos B \cdot x}}{\sin B}\right)\right) + \frac{1}{\sin B} \]

   15. distribute-neg-frac N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\cos B \cdot x\right)}{\sin B}} + \frac{1}{\sin B} \]

   16. div-add-rev N/A

       \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\cos B \cdot x\right)\right) + 1}{\sin B}} \]

   17. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\cos B \cdot x\right)\right) + 1}{\sin B}} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-x, \cos B, 1\right)}{\sin B}} \]

7. Taylor expanded in B around 0

   \[\leadsto \frac{\color{blue}{1 + -1 \cdot x}}{\sin B} \]
8. Step-by-step derivation

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\color{blue}{1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot x}}{\sin B} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1 - \color{blue}{1} \cdot x}{\sin B} \]

   3. *-lft-identity N/A

      \[\leadsto \frac{1 - \color{blue}{x}}{\sin B} \]

   4. lower--.f64 78.3

      \[\leadsto \frac{\color{blue}{1 - x}}{\sin B} \]

9. Applied rewrites 78.3%

   \[\leadsto \frac{\color{blue}{1 - x}}{\sin B} \]
10. Add Preprocessing

Alternative 6: 51.4% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (/ (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x) B))

C

double code(double B, double x) {
	return (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
}

Julia

function code(B, x)
	return Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B)
end

Wolfram

code[B_, x_] := N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}}{B} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + 1\right)} - x}{B} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{6} + \frac{1}{3} \cdot x\right) \cdot {B}^{2}} + 1\right) - x}{B} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{3} \cdot x, {B}^{2}, 1\right)} - x}{B} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot x + \frac{1}{6}}, {B}^{2}, 1\right) - x}{B} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3}, x, \frac{1}{6}\right)}, {B}^{2}, 1\right) - x}{B} \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{1}{6}\right), \color{blue}{B \cdot B}, 1\right) - x}{B} \]

   9. lower-*.f64 51.9

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), \color{blue}{B \cdot B}, 1\right) - x}{B} \]

5. Applied rewrites 51.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}} \]
6. Add Preprocessing

Alternative 7: 50.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-6} \lor \neg \left(x \leq 2.3 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -5.8e-6) (not (<= x 2.3e-13))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -5.8e-6) || !(x <= 2.3e-13)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5.8d-6)) .or. (.not. (x <= 2.3d-13))) then
        tmp = -x / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -5.8e-6) || !(x <= 2.3e-13)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -5.8e-6) or not (x <= 2.3e-13):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -5.8e-6) || !(x <= 2.3e-13))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -5.8e-6) || ~((x <= 2.3e-13)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -5.8e-6], N[Not[LessEqual[x, 2.3e-13]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8000000000000004e-6 or 2.29999999999999979e-13 < x

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

      2. lower--.f64 52.5

         \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

   5. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.7%

      \[\leadsto \frac{-x}{B} \]

   if -5.8000000000000004e-6 < x < 2.29999999999999979e-13

   1. Initial program 99.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

      2. lower--.f64 51.0

         \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

   5. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.0%

      \[\leadsto \frac{1}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-6} \lor \neg \left(x \leq 2.3 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.3% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (- 1.0 x) B))

C

double code(double B, double x) {
	return (1.0 - x) / B;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 - x) / b
end function

Java

public static double code(double B, double x) {
	return (1.0 - x) / B;
}

Python

def code(B, x):
	return (1.0 - x) / B

Julia

function code(B, x)
	return Float64(Float64(1.0 - x) / B)
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 - x) / B;
end

Wolfram

code[B_, x_] := N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   2. lower--.f64 51.8

      \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

5. Applied rewrites 51.8%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
6. Add Preprocessing

Alternative 9: 26.3% accurate, 19.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ 1.0 B))

C

double code(double B, double x) {
	return 1.0 / B;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double B, double x) {
	return 1.0 / B;
}

Python

def code(B, x):
	return 1.0 / B

Julia

function code(B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(B, x)
	tmp = 1.0 / B;
end

Wolfram

code[B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   2. lower--.f64 51.8

      \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

5. Applied rewrites 51.8%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{1}{B} \]
7. Step-by-step derivation

8. Applied rewrites 26.8%

   \[\leadsto \frac{1}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024340 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))