VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 7.9s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 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 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift-tan.f64 N/A

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

   9. tan-quot N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. clear-num N/A

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

   13. associate-/l* N/A

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

   14. lift-*.f64 N/A

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

   15. sub-div N/A

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

   16. lower-/.f64 N/A

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

   17. lower--.f64 99.7

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 3: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{\tan B}\\ \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.7:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) (tan B))))
   (if (<= x -1.2) t_0 (if (<= x 1.7) (- (/ 1.0 (sin B)) (/ x B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 - x) / tan(B);
	double tmp;
	if (x <= -1.2) {
		tmp = t_0;
	} else if (x <= 1.7) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - x) / tan(b)
    if (x <= (-1.2d0)) then
        tmp = t_0
    else if (x <= 1.7d0) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (1.0 - x) / Math.tan(B);
	double tmp;
	if (x <= -1.2) {
		tmp = t_0;
	} else if (x <= 1.7) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 - x) / math.tan(B)
	tmp = 0
	if x <= -1.2:
		tmp = t_0
	elif x <= 1.7:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 - x) / tan(B))
	tmp = 0.0
	if (x <= -1.2)
		tmp = t_0;
	elseif (x <= 1.7)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 - x) / tan(B);
	tmp = 0.0;
	if (x <= -1.2)
		tmp = t_0;
	elseif (x <= 1.7)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2], t$95$0, If[LessEqual[x, 1.7], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.19999999999999996 or 1.69999999999999996 < 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

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. 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-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-add N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. /-rgt-identity N/A

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

      11. lift-fma.f64 N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 99.1

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

   9. Applied rewrites 99.1%

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

   if -1.19999999999999996 < x < 1.69999999999999996

   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

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

      1. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{elif}\;x \leq 1.7:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{\tan B}\\ \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) (tan B))))
   (if (<= x -1.3) t_0 (if (<= x 1.0) (/ (- 1.0 x) (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 - x) / tan(B);
	double tmp;
	if (x <= -1.3) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - x) / tan(b)
    if (x <= (-1.3d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (1.0 - x) / Math.tan(B);
	double tmp;
	if (x <= -1.3) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 - x) / math.tan(B)
	tmp = 0
	if x <= -1.3:
		tmp = t_0
	elif x <= 1.0:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 - x) / tan(B))
	tmp = 0.0
	if (x <= -1.3)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 - x) / tan(B);
	tmp = 0.0;
	if (x <= -1.3)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3], t$95$0, If[LessEqual[x, 1.0], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000004 or 1 < 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

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. 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-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-add N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. /-rgt-identity N/A

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

      11. lift-fma.f64 N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 99.1

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

   9. Applied rewrites 99.1%

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

   if -1.30000000000000004 < x < 1

   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. un-div-inv N/A

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

      4. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-tan.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/l* N/A

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

      14. lift-*.f64 N/A

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

      15. sub-div N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 99.7

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. lower--.f64 98.5

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

   9. Applied rewrites 98.5%

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

Alternative 5: 76.3% 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 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift-tan.f64 N/A

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

   9. tan-quot N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. clear-num N/A

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

   13. associate-/l* N/A

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

   14. lift-*.f64 N/A

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

   15. sub-div N/A

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

   16. lower-/.f64 N/A

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

   17. lower--.f64 99.7

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in B around 0

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

   1. lower--.f64 74.8

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

9. Applied rewrites 74.8%

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

Alternative 6: 51.8% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 / b) - ((x / b) - ((0.3333333333333333d0 * b) * x))
end function

Java

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

Python

def code(B, x):
	return (1.0 / B) - ((x / B) - ((0.3333333333333333 * B) * x))

Julia

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

MATLAB

function tmp = code(B, x)
	tmp = (1.0 / B) - ((x / B) - ((0.3333333333333333 * B) * x));
end

Wolfram

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

Derivation

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

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

   1. metadata-eval N/A

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

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

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. div-sub N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-*r/ N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

   14. *-inverses N/A

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

   15. *-rgt-identity N/A

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

   16. *-commutative N/A

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

   17. associate-*l* N/A

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

5. Applied rewrites 57.1%

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

6. Taylor expanded in B around 0

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

   1. lower-/.f64 45.4

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

8. Applied rewrites 45.4%

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

9. Final simplification 45.4%

   \[\leadsto \frac{1}{B} - \left(\frac{x}{B} - \left(0.3333333333333333 \cdot B\right) \cdot x\right) \]
10. Add Preprocessing

Alternative 7: 51.5% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. rem-log-exp N/A

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

   5. pow-to-exp N/A

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

   6. inv-pow N/A

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

   7. lift-/.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-log.f64 50.6

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

6. Applied rewrites 50.6%

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

7. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{B \cdot \left(e^{\mathsf{neg}\left(\log B\right)} + \frac{1}{3} \cdot \left(B \cdot x\right)\right) - x}{B}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

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

9. Applied rewrites 45.0%

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

Alternative 8: 51.4% 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 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 44.9

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

5. Applied rewrites 44.9%

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

Alternative 9: 26.8% 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 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 44.9

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

5. Applied rewrites 44.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 20.9%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))