invcot (example 3.9)

Percentage Accurate: 6.5% → 99.5%

Time: 19.0s

Alternatives: 7

Speedup: 21.0×

Specification

\[-0.026 < x \land x < 0.026\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x, 0\right), 0\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), t\_0 \cdot 1.0973936899862826 \cdot 10^{-5}, 0.037037037037037035\right) \cdot \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), -0.007407407407407408, \mathsf{fma}\left(0.0004938271604938272, t\_0, 0.1111111111111111\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma x x 0.0) (fma x x 0.0) 0.0)))
   (*
    (fma (fma x x 0.0) (* t_0 1.0973936899862826e-5) 0.037037037037037035)
    (/
     x
     (fma
      (fma x x 0.0)
      -0.007407407407407408
      (fma 0.0004938271604938272 t_0 0.1111111111111111))))))

C

double code(double x) {
	double t_0 = fma(fma(x, x, 0.0), fma(x, x, 0.0), 0.0);
	return fma(fma(x, x, 0.0), (t_0 * 1.0973936899862826e-5), 0.037037037037037035) * (x / fma(fma(x, x, 0.0), -0.007407407407407408, fma(0.0004938271604938272, t_0, 0.1111111111111111)));
}

Julia

function code(x)
	t_0 = fma(fma(x, x, 0.0), fma(x, x, 0.0), 0.0)
	return Float64(fma(fma(x, x, 0.0), Float64(t_0 * 1.0973936899862826e-5), 0.037037037037037035) * Float64(x / fma(fma(x, x, 0.0), -0.007407407407407408, fma(0.0004938271604938272, t_0, 0.1111111111111111))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x + 0.0), $MachinePrecision] * N[(x * x + 0.0), $MachinePrecision] + 0.0), $MachinePrecision]}, N[(N[(N[(x * x + 0.0), $MachinePrecision] * N[(t$95$0 * 1.0973936899862826e-5), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] * N[(x / N[(N[(x * x + 0.0), $MachinePrecision] * -0.007407407407407408 + N[(0.0004938271604938272 * t$95$0 + 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.1%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {x}^{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{45} \cdot {x}^{2} + \frac{1}{3}\right)} \]

   3. unpow2 N/A

      \[\leadsto x \cdot \left(\frac{1}{45} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{3}\right) \]

   4. associate-*r* N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(\frac{1}{45} \cdot x\right) \cdot x} + \frac{1}{3}\right) \]

   5. *-commutative N/A

      \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(\frac{1}{45} \cdot x\right)} + \frac{1}{3}\right) \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{45} \cdot x, \frac{1}{3}\right)} \]

   7. +-rgt-identity N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{45} \cdot x + 0}, \frac{1}{3}\right) \]

   8. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{45}} + 0, \frac{1}{3}\right) \]

   9. accelerator-lowering-fma.f64 99.3

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.022222222222222223, 0\right)}, 0.3333333333333333\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.022222222222222223, 0\right), 0.3333333333333333\right)} \]

7. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1.0973936899862826 \cdot 10^{-5}, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x, 0\right), 0\right), 0\right), 0.037037037037037035\right) \cdot x}{\mathsf{fma}\left(0.0004938271604938272, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x, 0\right), 0\right), 0.1111111111111111\right) - 0.3333333333333333 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.022222222222222223, 0\right), 0\right)}} \]

9. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x, 0\right), 0\right) \cdot 1.0973936899862826 \cdot 10^{-5}, 0.037037037037037035\right) \cdot \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), -0.007407407407407408, \mathsf{fma}\left(0.0004938271604938272, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x, 0\right), 0\right), 0.1111111111111111\right)\right)}} \]
10. Add Preprocessing

Alternative 2: 99.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.00021164021164021165, 0.0021164021164021165\right), 0.022222222222222223\right), 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fma
   (* x x)
   (fma
    (* x x)
    (fma (* x x) 0.00021164021164021165 0.0021164021164021165)
    0.022222222222222223)
   0.3333333333333333)))

C

double code(double x) {
	return x * fma((x * x), fma((x * x), fma((x * x), 0.00021164021164021165, 0.0021164021164021165), 0.022222222222222223), 0.3333333333333333);
}

Julia

function code(x)
	return Float64(x * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), 0.00021164021164021165, 0.0021164021164021165), 0.022222222222222223), 0.3333333333333333))
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.00021164021164021165 + 0.0021164021164021165), $MachinePrecision] + 0.022222222222222223), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{45} + {x}^{2} \cdot \left(\frac{2}{945} + \frac{1}{4725} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{45} + {x}^{2} \cdot \left(\frac{2}{945} + \frac{1}{4725} \cdot {x}^{2}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{45} + {x}^{2} \cdot \left(\frac{2}{945} + \frac{1}{4725} \cdot {x}^{2}\right)\right) + \frac{1}{3}\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{45} + {x}^{2} \cdot \left(\frac{2}{945} + \frac{1}{4725} \cdot {x}^{2}\right), \frac{1}{3}\right)} \]

5. Simplified 99.5%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot 0.00021164021164021165, 0.0021164021164021165\right), 0.022222222222222223\right), 0.3333333333333333\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto x \cdot \color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{45} + {x}^{2} \cdot \left(\frac{2}{945} + \frac{1}{4725} \cdot {x}^{2}\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{45} + {x}^{2} \cdot \left(\frac{2}{945} + \frac{1}{4725} \cdot {x}^{2}\right)\right) + \frac{1}{3}\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{45} + {x}^{2} \cdot \left(\frac{2}{945} + \frac{1}{4725} \cdot {x}^{2}\right), \frac{1}{3}\right)} \]

   3. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{45} + {x}^{2} \cdot \left(\frac{2}{945} + \frac{1}{4725} \cdot {x}^{2}\right), \frac{1}{3}\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{45} + {x}^{2} \cdot \left(\frac{2}{945} + \frac{1}{4725} \cdot {x}^{2}\right), \frac{1}{3}\right) \]

   5. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2}{945} + \frac{1}{4725} \cdot {x}^{2}\right) + \frac{1}{45}}, \frac{1}{3}\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{945} + \frac{1}{4725} \cdot {x}^{2}, \frac{1}{45}\right)}, \frac{1}{3}\right) \]

   7. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{945} + \frac{1}{4725} \cdot {x}^{2}, \frac{1}{45}\right), \frac{1}{3}\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{945} + \frac{1}{4725} \cdot {x}^{2}, \frac{1}{45}\right), \frac{1}{3}\right) \]

   9. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{4725} \cdot {x}^{2} + \frac{2}{945}}, \frac{1}{45}\right), \frac{1}{3}\right) \]

   10. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{4725}} + \frac{2}{945}, \frac{1}{45}\right), \frac{1}{3}\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{4725}, \frac{2}{945}\right)}, \frac{1}{45}\right), \frac{1}{3}\right) \]

   12. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{4725}, \frac{2}{945}\right), \frac{1}{45}\right), \frac{1}{3}\right) \]

   13. *-lowering-*.f64 99.5

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.00021164021164021165, 0.0021164021164021165\right), 0.022222222222222223\right), 0.3333333333333333\right) \]

8. Simplified 99.5%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.00021164021164021165, 0.0021164021164021165\right), 0.022222222222222223\right), 0.3333333333333333\right)} \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.0021164021164021165, 0.022222222222222223\right), x, x \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* (* x x) (fma x (* x 0.0021164021164021165) 0.022222222222222223))
  x
  (* x 0.3333333333333333)))

C

double code(double x) {
	return fma(((x * x) * fma(x, (x * 0.0021164021164021165), 0.022222222222222223)), x, (x * 0.3333333333333333));
}

Julia

function code(x)
	return fma(Float64(Float64(x * x) * fma(x, Float64(x * 0.0021164021164021165), 0.022222222222222223)), x, Float64(x * 0.3333333333333333))
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.0021164021164021165), $MachinePrecision] + 0.022222222222222223), $MachinePrecision]), $MachinePrecision] * x + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right) + \frac{1}{3}\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right)} \]

   4. remove-double-neg N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   5. +-rgt-identity N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) + 0\right)}\right), \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   6. distribute-neg-in N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   7. remove-double-neg N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{{x}^{2}} + \left(\mathsf{neg}\left(0\right)\right), \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(0\right)\right), \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   9. metadata-eval N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x + \color{blue}{0}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   11. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{2}{945} \cdot {x}^{2} + \frac{1}{45}}, \frac{1}{3}\right) \]

   12. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{2}{945} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{45}, \frac{1}{3}\right) \]

   13. associate-*r* N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(\frac{2}{945} \cdot x\right) \cdot x} + \frac{1}{45}, \frac{1}{3}\right) \]

   14. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(\frac{2}{945} \cdot x\right)} + \frac{1}{45}, \frac{1}{3}\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, \frac{2}{945} \cdot x, \frac{1}{45}\right)}, \frac{1}{3}\right) \]

   16. +-rgt-identity N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\frac{2}{945} \cdot x + 0}, \frac{1}{45}\right), \frac{1}{3}\right) \]

   17. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{945}} + 0, \frac{1}{45}\right), \frac{1}{3}\right) \]

   18. accelerator-lowering-fma.f64 99.4

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.0021164021164021165, 0\right)}, 0.022222222222222223\right), 0.3333333333333333\right) \]

5. Simplified 99.4%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0021164021164021165, 0\right), 0.022222222222222223\right), 0.3333333333333333\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{945} + 0\right) + \frac{1}{45}\right)\right) \cdot x + \frac{1}{3} \cdot x} \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{945} + 0\right) + \frac{1}{45}\right), x, \frac{1}{3} \cdot x\right)} \]

   3. +-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{2}{945} + 0\right) + \frac{1}{45}\right), x, \frac{1}{3} \cdot x\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{2}{945} + 0\right) + \frac{1}{45}\right)\right)}, x, \frac{1}{3} \cdot x\right) \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{2}{945} + 0\right) + \frac{1}{45}\right)\right)}, x, \frac{1}{3} \cdot x\right) \]

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

      \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{2}{945} + 0\right) + \frac{1}{45}\right)\right)}, x, \frac{1}{3} \cdot x\right) \]

   7. +-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{2}{945}\right)} + \frac{1}{45}\right)\right), x, \frac{1}{3} \cdot x\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \frac{2}{945}} + \frac{1}{45}\right)\right), x, \frac{1}{3} \cdot x\right) \]

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

      \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{2}{945}, \frac{1}{45}\right)}\right), x, \frac{1}{3} \cdot x\right) \]

   10. +-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x + 0}, \frac{2}{945}, \frac{1}{45}\right)\right), x, \frac{1}{3} \cdot x\right) \]

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

       \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{2}{945}, \frac{1}{45}\right)\right), x, \frac{1}{3} \cdot x\right) \]

   12. *-lowering-*.f64 99.4

       \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0021164021164021165, 0.022222222222222223\right)\right), x, \color{blue}{0.3333333333333333 \cdot x}\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0021164021164021165, 0.022222222222222223\right)\right), x, 0.3333333333333333 \cdot x\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right)}, x, \frac{1}{3} \cdot x\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right)}, x, \frac{1}{3} \cdot x\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right), x, \frac{1}{3} \cdot x\right) \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right), x, \frac{1}{3} \cdot x\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{2}{945} \cdot {x}^{2} + \frac{1}{45}\right)}, x, \frac{1}{3} \cdot x\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{2}{945}} + \frac{1}{45}\right), x, \frac{1}{3} \cdot x\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{2}{945} + \frac{1}{45}\right), x, \frac{1}{3} \cdot x\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{2}{945}\right)} + \frac{1}{45}\right), x, \frac{1}{3} \cdot x\right) \]

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

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{2}{945}, \frac{1}{45}\right)}, x, \frac{1}{3} \cdot x\right) \]

   9. *-lowering-*.f64 99.4

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0021164021164021165}, 0.022222222222222223\right), x, 0.3333333333333333 \cdot x\right) \]

10. Simplified 99.4%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.0021164021164021165, 0.022222222222222223\right)}, x, 0.3333333333333333 \cdot x\right) \]

11. Final simplification 99.4%

    \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.0021164021164021165, 0.022222222222222223\right), x, x \cdot 0.3333333333333333\right) \]
12. Add Preprocessing

Alternative 4: 99.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0021164021164021165, 0.022222222222222223\right), 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fma
   (* x x)
   (fma x (* x 0.0021164021164021165) 0.022222222222222223)
   0.3333333333333333)))

C

double code(double x) {
	return x * fma((x * x), fma(x, (x * 0.0021164021164021165), 0.022222222222222223), 0.3333333333333333);
}

Julia

function code(x)
	return Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.0021164021164021165), 0.022222222222222223), 0.3333333333333333))
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.0021164021164021165), $MachinePrecision] + 0.022222222222222223), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right) + \frac{1}{3}\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right)} \]

   4. remove-double-neg N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   5. +-rgt-identity N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) + 0\right)}\right), \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   6. distribute-neg-in N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   7. remove-double-neg N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{{x}^{2}} + \left(\mathsf{neg}\left(0\right)\right), \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(0\right)\right), \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   9. metadata-eval N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x + \color{blue}{0}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   11. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{2}{945} \cdot {x}^{2} + \frac{1}{45}}, \frac{1}{3}\right) \]

   12. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{2}{945} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{45}, \frac{1}{3}\right) \]

   13. associate-*r* N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(\frac{2}{945} \cdot x\right) \cdot x} + \frac{1}{45}, \frac{1}{3}\right) \]

   14. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(\frac{2}{945} \cdot x\right)} + \frac{1}{45}, \frac{1}{3}\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, \frac{2}{945} \cdot x, \frac{1}{45}\right)}, \frac{1}{3}\right) \]

   16. +-rgt-identity N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\frac{2}{945} \cdot x + 0}, \frac{1}{45}\right), \frac{1}{3}\right) \]

   17. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{945}} + 0, \frac{1}{45}\right), \frac{1}{3}\right) \]

   18. accelerator-lowering-fma.f64 99.4

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.0021164021164021165, 0\right)}, 0.022222222222222223\right), 0.3333333333333333\right) \]

5. Simplified 99.4%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0021164021164021165, 0\right), 0.022222222222222223\right), 0.3333333333333333\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto x \cdot \color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right) + \frac{1}{3}\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right)} \]

   3. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   5. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2}{945} \cdot {x}^{2} + \frac{1}{45}}, \frac{1}{3}\right) \]

   6. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{2}{945}} + \frac{1}{45}, \frac{1}{3}\right) \]

   7. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{2}{945} + \frac{1}{45}, \frac{1}{3}\right) \]

   8. associate-*l* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{2}{945}\right)} + \frac{1}{45}, \frac{1}{3}\right) \]

   9. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{2}{945} \cdot x\right)} + \frac{1}{45}, \frac{1}{3}\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{2}{945} \cdot x, \frac{1}{45}\right)}, \frac{1}{3}\right) \]

   11. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{945}}, \frac{1}{45}\right), \frac{1}{3}\right) \]

   12. *-lowering-*.f64 99.4

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0021164021164021165}, 0.022222222222222223\right), 0.3333333333333333\right) \]

8. Simplified 99.4%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0021164021164021165, 0.022222222222222223\right), 0.3333333333333333\right)} \]
9. Add Preprocessing

Alternative 5: 99.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.022222222222222223, x, x \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* (* x x) 0.022222222222222223) x (* x 0.3333333333333333)))

C

double code(double x) {
	return fma(((x * x) * 0.022222222222222223), x, (x * 0.3333333333333333));
}

Julia

function code(x)
	return fma(Float64(Float64(x * x) * 0.022222222222222223), x, Float64(x * 0.3333333333333333))
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 0.022222222222222223), $MachinePrecision] * x + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {x}^{2}\right) + \frac{1}{3}\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right)} \]

   4. remove-double-neg N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   5. +-rgt-identity N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) + 0\right)}\right), \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   6. distribute-neg-in N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   7. remove-double-neg N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{{x}^{2}} + \left(\mathsf{neg}\left(0\right)\right), \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(0\right)\right), \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   9. metadata-eval N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x + \color{blue}{0}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{45} + \frac{2}{945} \cdot {x}^{2}, \frac{1}{3}\right) \]

   11. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{2}{945} \cdot {x}^{2} + \frac{1}{45}}, \frac{1}{3}\right) \]

   12. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{2}{945} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{45}, \frac{1}{3}\right) \]

   13. associate-*r* N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(\frac{2}{945} \cdot x\right) \cdot x} + \frac{1}{45}, \frac{1}{3}\right) \]

   14. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(\frac{2}{945} \cdot x\right)} + \frac{1}{45}, \frac{1}{3}\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, \frac{2}{945} \cdot x, \frac{1}{45}\right)}, \frac{1}{3}\right) \]

   16. +-rgt-identity N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\frac{2}{945} \cdot x + 0}, \frac{1}{45}\right), \frac{1}{3}\right) \]

   17. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{945}} + 0, \frac{1}{45}\right), \frac{1}{3}\right) \]

   18. accelerator-lowering-fma.f64 99.4

       \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.0021164021164021165, 0\right)}, 0.022222222222222223\right), 0.3333333333333333\right) \]

5. Simplified 99.4%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0021164021164021165, 0\right), 0.022222222222222223\right), 0.3333333333333333\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{945} + 0\right) + \frac{1}{45}\right)\right) \cdot x + \frac{1}{3} \cdot x} \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{945} + 0\right) + \frac{1}{45}\right), x, \frac{1}{3} \cdot x\right)} \]

   3. +-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{2}{945} + 0\right) + \frac{1}{45}\right), x, \frac{1}{3} \cdot x\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{2}{945} + 0\right) + \frac{1}{45}\right)\right)}, x, \frac{1}{3} \cdot x\right) \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{2}{945} + 0\right) + \frac{1}{45}\right)\right)}, x, \frac{1}{3} \cdot x\right) \]

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

      \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{2}{945} + 0\right) + \frac{1}{45}\right)\right)}, x, \frac{1}{3} \cdot x\right) \]

   7. +-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{2}{945}\right)} + \frac{1}{45}\right)\right), x, \frac{1}{3} \cdot x\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \frac{2}{945}} + \frac{1}{45}\right)\right), x, \frac{1}{3} \cdot x\right) \]

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

      \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{2}{945}, \frac{1}{45}\right)}\right), x, \frac{1}{3} \cdot x\right) \]

   10. +-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x + 0}, \frac{2}{945}, \frac{1}{45}\right)\right), x, \frac{1}{3} \cdot x\right) \]

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

       \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{2}{945}, \frac{1}{45}\right)\right), x, \frac{1}{3} \cdot x\right) \]

   12. *-lowering-*.f64 99.4

       \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0021164021164021165, 0.022222222222222223\right)\right), x, \color{blue}{0.3333333333333333 \cdot x}\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0021164021164021165, 0.022222222222222223\right)\right), x, 0.3333333333333333 \cdot x\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{45} \cdot {x}^{2}}, x, \frac{1}{3} \cdot x\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \frac{1}{45}}, x, \frac{1}{3} \cdot x\right) \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \frac{1}{45}}, x, \frac{1}{3} \cdot x\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{45}, x, \frac{1}{3} \cdot x\right) \]

   4. *-lowering-*.f64 99.3

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot 0.022222222222222223, x, 0.3333333333333333 \cdot x\right) \]

10. Simplified 99.3%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right) \cdot 0.022222222222222223}, x, 0.3333333333333333 \cdot x\right) \]

11. Final simplification 99.3%

    \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.022222222222222223, x, x \cdot 0.3333333333333333\right) \]
12. Add Preprocessing

Alternative 6: 99.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x, x \cdot 0.022222222222222223, 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (fma x (* x 0.022222222222222223) 0.3333333333333333)))

C

double code(double x) {
	return x * fma(x, (x * 0.022222222222222223), 0.3333333333333333);
}

Julia

function code(x)
	return Float64(x * fma(x, Float64(x * 0.022222222222222223), 0.3333333333333333))
end

Wolfram

code[x_] := N[(x * N[(x * N[(x * 0.022222222222222223), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {x}^{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{45} \cdot {x}^{2} + \frac{1}{3}\right)} \]

   3. unpow2 N/A

      \[\leadsto x \cdot \left(\frac{1}{45} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{3}\right) \]

   4. associate-*r* N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(\frac{1}{45} \cdot x\right) \cdot x} + \frac{1}{3}\right) \]

   5. *-commutative N/A

      \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(\frac{1}{45} \cdot x\right)} + \frac{1}{3}\right) \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{45} \cdot x, \frac{1}{3}\right)} \]

   7. +-rgt-identity N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{45} \cdot x + 0}, \frac{1}{3}\right) \]

   8. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{45}} + 0, \frac{1}{3}\right) \]

   9. accelerator-lowering-fma.f64 99.3

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.022222222222222223, 0\right)}, 0.3333333333333333\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.022222222222222223, 0\right), 0.3333333333333333\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{45}}, \frac{1}{3}\right) \]

   2. *-lowering-*.f64 99.3

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.022222222222222223}, 0.3333333333333333\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.022222222222222223}, 0.3333333333333333\right) \]
8. Add Preprocessing

Alternative 7: 98.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.3333333333333333))

C

double code(double x) {
	return x * 0.3333333333333333;
}

Fortran

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

Java

public static double code(double x) {
	return x * 0.3333333333333333;
}

Python

def code(x):
	return x * 0.3333333333333333

Julia

function code(x)
	return Float64(x * 0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.3333333333333333;
end

Wolfram

code[x_] := N[(x * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 7.1%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot x} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 98.4

      \[\leadsto \color{blue}{0.3333333333333333 \cdot x} \]

5. Simplified 98.4%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot x} \]

6. Final simplification 98.4%

   \[\leadsto x \cdot 0.3333333333333333 \]
7. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| < 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (< (fabs x) 0.026)
   (* (/ x 3.0) (+ 1.0 (/ (* x x) 15.0)))
   (- (/ 1.0 x) (/ 1.0 (tan x)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) < 0.026) {
		tmp = (x / 3.0) * (1.0 + ((x * x) / 15.0));
	} else {
		tmp = (1.0 / x) - (1.0 / tan(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) < 0.026d0) then
        tmp = (x / 3.0d0) * (1.0d0 + ((x * x) / 15.0d0))
    else
        tmp = (1.0d0 / x) - (1.0d0 / tan(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) < 0.026) {
		tmp = (x / 3.0) * (1.0 + ((x * x) / 15.0));
	} else {
		tmp = (1.0 / x) - (1.0 / Math.tan(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) < 0.026:
		tmp = (x / 3.0) * (1.0 + ((x * x) / 15.0))
	else:
		tmp = (1.0 / x) - (1.0 / math.tan(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) < 0.026)
		tmp = Float64(Float64(x / 3.0) * Float64(1.0 + Float64(Float64(x * x) / 15.0)));
	else
		tmp = Float64(Float64(1.0 / x) - Float64(1.0 / tan(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) < 0.026)
		tmp = (x / 3.0) * (1.0 + ((x * x) / 15.0));
	else
		tmp = (1.0 / x) - (1.0 / tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Less[N[Abs[x], $MachinePrecision], 0.026], N[(N[(x / 3.0), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] / 15.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024194 
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :pre (and (< -0.026 x) (< x 0.026))

  :alt
  (! :herbie-platform default (if (< (fabs x) 13/500) (* (/ x 3) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x)))))

  (- (/ 1.0 x) (/ 1.0 (tan x))))