Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 12.5s

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	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) ** 2.0d0
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. div-sub 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. add-sqr-sqrt 99.2%

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

   4. prod-diff 99.3%

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

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}} + \mathsf{fma}\left(-\sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}\right)} \]
5. Step-by-step derivation

   1. fma-undefine 99.1%

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

6. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}}} \]

7. Final simplification 99.5%

   \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}} \]
8. Add Preprocessing

Alternative 2: 54.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\tan x}^{2}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 + (tan(x) ** 2.0d0))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 + Math.pow(Math.tan(x), 2.0));
}

Python

def code(x):
	return 1.0 / (1.0 + math.pow(math.tan(x), 2.0))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 + (tan(x) ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 + (tan(x) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. div-sub 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. add-sqr-sqrt 99.2%

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

   4. prod-diff 99.3%

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

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}} + \mathsf{fma}\left(-\sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}\right)} \]
5. Step-by-step derivation

   1. fma-undefine 99.1%

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

6. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}}} \]

7. Taylor expanded in x around 0 58.1%

   \[\leadsto \frac{\color{blue}{1}}{1 + {\tan x}^{2}} \]

8. Final simplification 58.1%

   \[\leadsto \frac{1}{1 + {\tan x}^{2}} \]
9. Add Preprocessing

Alternative 3: 57.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 - {\tan x}^{2}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 - (tan(x) ** 2.0d0))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 - Math.pow(Math.tan(x), 2.0));
}

Python

def code(x):
	return 1.0 / (1.0 - math.pow(math.tan(x), 2.0))

Julia

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

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 - (tan(x) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. div-sub 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. add-sqr-sqrt 99.2%

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

   4. prod-diff 99.3%

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

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}} + \mathsf{fma}\left(-\sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}\right)} \]
5. Step-by-step derivation

   1. fma-undefine 99.1%

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

6. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}}} \]

7. Taylor expanded in x around 0 58.1%

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

   1. frac-2neg 58.1%

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

   2. metadata-eval 58.1%

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

   3. div-inv 58.1%

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

   4. +-commutative 58.1%

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

   5. distribute-neg-in 58.1%

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

   6. pow2 58.1%

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

   7. distribute-rgt-neg-in 58.1%

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

   8. pow1 58.1%

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

   9. metadata-eval 58.1%

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

   10. sqrt-pow1 60.0%

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

   11. add-sqr-sqrt 30.3%

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

   12. sqrt-unprod 61.2%

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

   13. sqr-neg 61.2%

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

   14. pow2 61.2%

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

   15. add-sqr-sqrt 61.2%

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

   16. metadata-eval 61.2%

       \[\leadsto -1 \cdot \frac{1}{{\tan x}^{2} + \color{blue}{-1}} \]

9. Applied egg-rr 61.2%

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

    1. mul-1-neg 61.2%

       \[\leadsto \color{blue}{-\frac{1}{{\tan x}^{2} + -1}} \]

    2. distribute-neg-frac2 61.2%

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

    3. +-commutative 61.2%

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

    4. distribute-neg-in 61.2%

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

    5. metadata-eval 61.2%

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

    6. sub-neg 61.2%

       \[\leadsto \frac{1}{\color{blue}{1 - {\tan x}^{2}}} \]

11. Simplified 61.2%

    \[\leadsto \color{blue}{\frac{1}{1 - {\tan x}^{2}}} \]

12. Final simplification 61.2%

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

Alternative 4: 50.3% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (pow (hypot 1.0 x) -2.0))

C

double code(double x) {
	return pow(hypot(1.0, x), -2.0);
}

Java

public static double code(double x) {
	return Math.pow(Math.hypot(1.0, x), -2.0);
}

Python

def code(x):
	return math.pow(math.hypot(1.0, x), -2.0)

Julia

function code(x)
	return hypot(1.0, x) ^ -2.0
end

MATLAB

function tmp = code(x)
	tmp = hypot(1.0, x) ^ -2.0;
end

Wolfram

code[x_] := N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. div-sub 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. add-sqr-sqrt 99.2%

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

   4. prod-diff 99.3%

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

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}} + \mathsf{fma}\left(-\sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}\right)} \]
5. Step-by-step derivation

   1. fma-undefine 99.1%

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

6. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}}} \]

7. Taylor expanded in x around 0 58.1%

   \[\leadsto \frac{\color{blue}{1}}{1 + {\tan x}^{2}} \]

8. Taylor expanded in x around 0 54.4%

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

   1. +-commutative 54.4%

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} + 1}} \]

   2. unpow2 54.4%

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

   3. fma-define 54.4%

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

10. Simplified 54.4%

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

    1. inv-pow 54.4%

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

    2. add-sqr-sqrt 54.4%

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

    3. unpow-prod-down 54.4%

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

12. Applied egg-rr 54.4%

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

    1. pow-sqr 54.4%

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

    2. fma-undefine 54.4%

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

    3. unpow2 54.4%

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

    4. +-commutative 54.4%

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

    5. unpow2 54.4%

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

    6. hypot-1-def 54.4%

       \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(1, x\right)\right)}}^{\left(2 \cdot -1\right)} \]

    7. metadata-eval 54.4%

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

14. Simplified 54.4%

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

15. Final simplification 54.4%

    \[\leadsto {\left(\mathsf{hypot}\left(1, x\right)\right)}^{-2} \]
16. Add Preprocessing

Alternative 5: 50.3% accurate, 3.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (fma x x 1.0)))

C

double code(double x) {
	return 1.0 / fma(x, x, 1.0);
}

Julia

function code(x)
	return Float64(1.0 / fma(x, x, 1.0))
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. div-sub 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. add-sqr-sqrt 99.2%

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

   4. prod-diff 99.3%

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

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}} + \mathsf{fma}\left(-\sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}\right)} \]
5. Step-by-step derivation

   1. fma-undefine 99.1%

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

6. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}}} \]

7. Taylor expanded in x around 0 58.1%

   \[\leadsto \frac{\color{blue}{1}}{1 + {\tan x}^{2}} \]

8. Taylor expanded in x around 0 54.4%

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

   1. +-commutative 54.4%

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} + 1}} \]

   2. unpow2 54.4%

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

   3. fma-define 54.4%

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

10. Simplified 54.4%

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

11. Final simplification 54.4%

    \[\leadsto \frac{1}{\mathsf{fma}\left(x, x, 1\right)} \]
12. Add Preprocessing

Alternative 6: 3.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (pow x 2.0)))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (x ** 2.0d0)
end function

Java

public static double code(double x) {
	return 1.0 / Math.pow(x, 2.0);
}

Python

def code(x):
	return 1.0 / math.pow(x, 2.0)

Julia

function code(x)
	return Float64(1.0 / (x ^ 2.0))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. div-sub 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. add-sqr-sqrt 99.2%

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

   4. prod-diff 99.3%

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

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}} + \mathsf{fma}\left(-\sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}\right)} \]
5. Step-by-step derivation

   1. fma-undefine 99.1%

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

6. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}}} \]

7. Taylor expanded in x around 0 58.1%

   \[\leadsto \frac{\color{blue}{1}}{1 + {\tan x}^{2}} \]

8. Taylor expanded in x around 0 54.4%

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

   1. +-commutative 54.4%

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} + 1}} \]

   2. unpow2 54.4%

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

   3. fma-define 54.4%

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

10. Simplified 54.4%

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

11. Taylor expanded in x around inf 4.0%

    \[\leadsto \color{blue}{\frac{1}{{x}^{2}}} \]

12. Final simplification 4.0%

    \[\leadsto \frac{1}{{x}^{2}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))