VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.7%

Time: 7.1s

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: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double B, double x) {
	return (1.0 / sin(B)) - ((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 / sin(b)) - ((cos(b) * x) / sin(b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-sin.f64 N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \frac{1}{\sin B} - \frac{\cos B \cdot x}{\sin B} \]
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 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-sin.f64 N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. div-inv N/A

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

   7. lift-/.f64 N/A

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

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

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. div-inv N/A

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

   12. sub-div N/A

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

   13. lower-/.f64 N/A

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

   14. lower--.f64 99.8

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 3: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -0.000135:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -0.000135) t_0 (if (<= x 5.2e-6) (/ 1.0 (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -0.000135) {
		tmp = t_0;
	} else if (x <= 5.2e-6) {
		tmp = 1.0 / 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 / b) - (x / tan(b))
    if (x <= (-0.000135d0)) then
        tmp = t_0
    else if (x <= 5.2d-6) then
        tmp = 1.0d0 / 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 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -0.000135) {
		tmp = t_0;
	} else if (x <= 5.2e-6) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -0.000135:
		tmp = t_0
	elif x <= 5.2e-6:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -0.000135)
		tmp = t_0;
	elseif (x <= 5.2e-6)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -0.000135)
		tmp = t_0;
	elseif (x <= 5.2e-6)
		tmp = 1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.000135], t$95$0, If[LessEqual[x, 5.2e-6], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000002e-4 or 5.20000000000000019e-6 < 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

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. clear-num N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. tan-quot N/A

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

      11. lift-tan.f64 N/A

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

      12. div-inv N/A

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

      13. lift-/.f64 N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 99.0

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

   9. Applied rewrites 99.0%

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

   if -1.35000000000000002e-4 < x < 5.20000000000000019e-6

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 4: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 32000000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= x -3.5e+32) t_0 (if (<= x 32000000.0) (/ (- 1.0 x) (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -3.5e+32) {
		tmp = t_0;
	} else if (x <= 32000000.0) {
		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) / b) - (x / tan(b))
    if (x <= (-3.5d+32)) then
        tmp = t_0
    else if (x <= 32000000.0d0) 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 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -3.5e+32) {
		tmp = t_0;
	} else if (x <= 32000000.0) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -3.5e+32:
		tmp = t_0
	elif x <= 32000000.0:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -3.5e+32)
		tmp = t_0;
	elseif (x <= 32000000.0)
		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 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -3.5e+32)
		tmp = t_0;
	elseif (x <= 32000000.0)
		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 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.5e+32], t$95$0, If[LessEqual[x, 32000000.0], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5000000000000001e32 or 3.2e7 < x

   1. Initial program 99.6%

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if -3.5000000000000001e32 < x < 3.2e7

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

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

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 99.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in B around 0

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

      1. lower--.f64 97.8

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

   9. Applied rewrites 97.8%

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

Alternative 5: 62.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 7 \cdot 10^{-12}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 7e-12) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 7e-12) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / 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 (b <= 7d-12) then
        tmp = (1.0d0 - x) / b
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 7e-12) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 7e-12:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 7e-12)
		tmp = Float64(Float64(1.0 - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 7e-12)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 7e-12], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 7.0000000000000001e-12

   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 63.2

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

   5. Applied rewrites 63.2%

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

   if 7.0000000000000001e-12 < B

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 45.6

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

   5. Applied rewrites 45.6%

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

Alternative 6: 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

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-sin.f64 N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. div-inv N/A

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

   7. lift-/.f64 N/A

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

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

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. div-inv N/A

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

   12. sub-div N/A

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

   13. lower-/.f64 N/A

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

   14. lower--.f64 99.8

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in B around 0

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

   1. lower--.f64 73.3

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

9. Applied rewrites 73.3%

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

Alternative 7: 50.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -100:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= x -100.0) t_0 (if (<= x 1.16e-17) (/ 1.0 B) t_0))))

C

double code(double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -100.0) {
		tmp = t_0;
	} else if (x <= 1.16e-17) {
		tmp = 1.0 / 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 = -x / b
    if (x <= (-100.0d0)) then
        tmp = t_0
    else if (x <= 1.16d-17) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -100.0) {
		tmp = t_0;
	} else if (x <= 1.16e-17) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -x / B
	tmp = 0
	if x <= -100.0:
		tmp = t_0
	elif x <= 1.16e-17:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(-x) / B)
	tmp = 0.0
	if (x <= -100.0)
		tmp = t_0;
	elseif (x <= 1.16e-17)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = -x / B;
	tmp = 0.0;
	if (x <= -100.0)
		tmp = t_0;
	elseif (x <= 1.16e-17)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[((-x) / B), $MachinePrecision]}, If[LessEqual[x, -100.0], t$95$0, If[LessEqual[x, 1.16e-17], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -100 or 1.16e-17 < 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 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 47.4%

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

   if -100 < x < 1.16e-17

   1. Initial program 99.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 53.4

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.7%

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

Alternative 8: 51.1% 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 50.7

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

5. Applied rewrites 50.7%

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

Alternative 9: 27.0% 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 50.7

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

5. Applied rewrites 50.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 24.3%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))