VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 7.7s

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 99.7%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := t\_0 - \frac{1}{\tan B} \cdot x\\ t_2 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+17}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 500:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B)))
        (t_1 (- t_0 (* (/ 1.0 (tan B)) x)))
        (t_2 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= t_1 -4e+17) t_2 (if (<= t_1 500.0) t_0 t_2))))

C

double code(double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = t_0 - ((1.0 / tan(B)) * x);
	double t_2 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (t_1 <= -4e+17) {
		tmp = t_2;
	} else if (t_1 <= 500.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = t_0 - ((1.0d0 / tan(b)) * x)
    t_2 = (1.0d0 / b) - (x / tan(b))
    if (t_1 <= (-4d+17)) then
        tmp = t_2
    else if (t_1 <= 500.0d0) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = t_0 - ((1.0 / Math.tan(B)) * x);
	double t_2 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (t_1 <= -4e+17) {
		tmp = t_2;
	} else if (t_1 <= 500.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = t_0 - ((1.0 / math.tan(B)) * x)
	t_2 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if t_1 <= -4e+17:
		tmp = t_2
	elif t_1 <= 500.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(t_0 - Float64(Float64(1.0 / tan(B)) * x))
	t_2 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_1 <= -4e+17)
		tmp = t_2;
	elseif (t_1 <= 500.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = 1.0 / sin(B);
	t_1 = t_0 - ((1.0 / tan(B)) * x);
	t_2 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (t_1 <= -4e+17)
		tmp = t_2;
	elseif (t_1 <= 500.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+17], t$95$2, If[LessEqual[t$95$1, 500.0], t$95$0, t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < -4e17 or 500 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B)))

   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-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -4e17 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 500

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 96.0

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

   5. Applied rewrites 96.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sin B} - \frac{1}{\tan B} \cdot x \leq -4 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;\frac{1}{\sin B} - \frac{1}{\tan B} \cdot x \leq 500:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% 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-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-/.f64 N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-tan.f64 N/A

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

   16. tan-quot N/A

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

   17. lift-sin.f64 N/A

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

   18. lift-cos.f64 N/A

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

   19. clear-num N/A

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

   20. associate-/l* N/A

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

   21. lift-*.f64 N/A

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

6. Applied rewrites 99.7%

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

Alternative 4: 63.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.0035:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.0035)
   (/ (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x) B)
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.0035) {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.0035)
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.0035], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.00350000000000000007

   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

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   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 68.8

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

   5. Applied rewrites 68.8%

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

   if 0.00350000000000000007 < 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 57.8

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

   5. Applied rewrites 57.8%

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

Alternative 5: 77.0% 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-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-add N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. *-rgt-identity N/A

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

   10. lower-fma.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in B around 0

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

   1. mul-1-neg N/A

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

   2. sub-neg N/A

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

   3. lower--.f64 80.8

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

9. Applied rewrites 80.8%

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

Alternative 6: 51.7% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[B_, x_] := N[(N[(N[((-x) * B + B), $MachinePrecision] / B), $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 52.7

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

5. Applied rewrites 52.7%

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

7. Applied rewrites 36.6%

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

9. Applied rewrites 52.8%

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

Alternative 7: 50.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= x -1.0) t_0 (if (<= x 2.8e-10) (/ 1.0 B) t_0))))

C

double code(double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 2.8e-10) {
		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 <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 2.8d-10) 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 <= -1.0) {
		tmp = t_0;
	} else if (x <= 2.8e-10) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -x / B
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 2.8e-10:
		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 <= -1.0)
		tmp = t_0;
	elseif (x <= 2.8e-10)
		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 <= -1.0)
		tmp = t_0;
	elseif (x <= 2.8e-10)
		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, -1.0], t$95$0, If[LessEqual[x, 2.8e-10], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 2.80000000000000015e-10 < x

   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 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.3%

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

   if -1 < x < 2.80000000000000015e-10

   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 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.2%

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

Alternative 8: 51.6% 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 52.7

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

5. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
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 52.7

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

5. Applied rewrites 52.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 27.4%

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

Reproduce

herbie shell --seed 2024251 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))