Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 8.9s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - {\tan x}^{2}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- 1.0 (pow (tan x) 2.0)) (fma (tan x) (tan x) 1.0)))

C

double code(double x) {
	return (1.0 - pow(tan(x), 2.0)) / fma(tan(x), tan(x), 1.0);
}

Julia

function code(x)
	return Float64(Float64(1.0 - (tan(x) ^ 2.0)) / fma(tan(x), tan(x), 1.0))
end

Wolfram

code[x_] := N[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 99.6

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

6. Applied rewrites 99.6%

   \[\leadsto \frac{1 - \color{blue}{{\tan x}^{2}}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
7. Add Preprocessing

Alternative 2: 60.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \mathbf{if}\;\tan x \cdot \tan x \leq 0.65:\\ \;\;\;\;\frac{1}{\frac{1 + t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right), -1, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (if (<= (* (tan x) (tan x)) 0.65)
     (/ 1.0 (/ (+ 1.0 t_0) (fma (fma (cos (+ x x)) -0.5 0.5) (- 1.0) 1.0)))
     (/ (- 1.0 t_0) 1.0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	double tmp;
	if ((tan(x) * tan(x)) <= 0.65) {
		tmp = 1.0 / ((1.0 + t_0) / fma(fma(cos((x + x)), -0.5, 0.5), -1.0, 1.0));
	} else {
		tmp = (1.0 - t_0) / 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 0.65)
		tmp = Float64(1.0 / Float64(Float64(1.0 + t_0) / fma(fma(cos(Float64(x + x)), -0.5, 0.5), Float64(-1.0), 1.0)));
	else
		tmp = Float64(Float64(1.0 - t_0) / 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 0.65], N[(1.0 / N[(N[(1.0 + t$95$0), $MachinePrecision] / N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * (-1.0) + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 0.650000000000000022

   1. Initial program 99.7%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. div-inv N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. div-inv N/A

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

      8. swap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. sqr-sin-a N/A

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

      11. lower--.f64 N/A

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

      12. cos-2 N/A

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

      13. cos-sum N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. inv-pow N/A

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

      18. inv-pow N/A

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

      19. pow-prod-down N/A

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

      20. inv-pow N/A

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

      21. lower-/.f64 N/A

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

      22. sqr-cos-a N/A

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

      23. lower-+.f64 N/A

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

      24. cos-2 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 78.5%

      \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \color{blue}{1}}{1 + \tan x \cdot \tan x} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-tan.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lower-+.f64 78.5

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

   9. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + {\tan x}^{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right), -1, 1\right)}}} \]

   if 0.650000000000000022 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 98.9%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 98.9

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

   4. Applied rewrites 98.9%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-pow.f64 98.9

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1 - {\tan x}^{2}}{\color{blue}{1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 16.6%

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

Reproduce

herbie shell --seed 2024229 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))