Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 12.7s

Alternatives: 7

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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. sub-neg 99.4%

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

   2. +-commutative 99.4%

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

   3. distribute-rgt-neg-in 99.4%

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

   4. fma-def 99.5%

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

3. Applied egg-rr 99.5%

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

   1. add-log-exp 97.9%

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

   2. *-un-lft-identity 97.9%

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

   3. log-prod 97.9%

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

   4. metadata-eval 97.9%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

5. Applied egg-rr 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \color{blue}{\left(0 + {\tan x}^{2}\right)}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 60.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1:\\ \;\;\;\;{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;e^{\log 2 - {x}^{2}} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.0)
   (pow (hypot 1.0 (tan x)) -2.0)
   (+ (exp (- (log 2.0) (pow x 2.0))) -1.0)))

C

double code(double x) {
	double tmp;
	if ((tan(x) * tan(x)) <= 1.0) {
		tmp = pow(hypot(1.0, tan(x)), -2.0);
	} else {
		tmp = exp((log(2.0) - pow(x, 2.0))) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((Math.tan(x) * Math.tan(x)) <= 1.0) {
		tmp = Math.pow(Math.hypot(1.0, Math.tan(x)), -2.0);
	} else {
		tmp = Math.exp((Math.log(2.0) - Math.pow(x, 2.0))) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.tan(x) * math.tan(x)) <= 1.0:
		tmp = math.pow(math.hypot(1.0, math.tan(x)), -2.0)
	else:
		tmp = math.exp((math.log(2.0) - math.pow(x, 2.0))) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 1.0)
		tmp = hypot(1.0, tan(x)) ^ -2.0;
	else
		tmp = Float64(exp(Float64(log(2.0) - (x ^ 2.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((tan(x) * tan(x)) <= 1.0)
		tmp = hypot(1.0, tan(x)) ^ -2.0;
	else
		tmp = exp((log(2.0) - (x ^ 2.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 1.0], N[Power[N[Sqrt[1.0 ^ 2 + N[Tan[x], $MachinePrecision] ^ 2], $MachinePrecision], -2.0], $MachinePrecision], N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 1

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. add-log-exp 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. log-prod 99.6%

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

      4. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

      5. add-log-exp 99.6%

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

      6. pow2 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \color{blue}{\left(0 + {\tan x}^{2}\right)}} \]
   6. Step-by-step derivation

      1. +-lft-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 70.6%

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

      1. expm1-log1p-u 70.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + {\tan x}^{2}}\right)\right)} \]

      2. expm1-udef 70.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{1 + {\tan x}^{2}}\right)} - 1} \]

      3. pow1 70.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{\left(1 + {\tan x}^{2}\right)}^{1}}}\right)} - 1 \]

      4. metadata-eval 70.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{{\left(1 + {\tan x}^{2}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}}\right)} - 1 \]

      5. sqrt-pow2 70.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{\left(\sqrt{1 + {\tan x}^{2}}\right)}^{2}}}\right)} - 1 \]

      6. metadata-eval 70.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{{\left(\sqrt{\color{blue}{1 \cdot 1} + {\tan x}^{2}}\right)}^{2}}\right)} - 1 \]

      7. unpow2 70.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{{\left(\sqrt{1 \cdot 1 + \color{blue}{\tan x \cdot \tan x}}\right)}^{2}}\right)} - 1 \]

      8. hypot-udef 70.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{{\color{blue}{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}}^{2}}\right)} - 1 \]

      9. pow-flip 70.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{\left(-2\right)}}\right)} - 1 \]

      10. metadata-eval 70.6%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{\color{blue}{-2}}\right)} - 1 \]

   10. Applied egg-rr 70.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 70.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2}\right)\right)} \]

       2. expm1-log1p 70.6%

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

   12. Simplified 70.6%

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

   if 1 < (*.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. Step-by-step derivation

      1. expm1-log1p-u 98.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\right)\right)} \]

      2. expm1-udef 99.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\right)} - 1} \]

      3. pow2 99.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1 - \color{blue}{{\tan x}^{2}}}{1 + \tan x \cdot \tan x}\right)} - 1 \]

      4. add-sqr-sqrt 98.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1 - {\tan x}^{2}}{\color{blue}{\sqrt{1 + \tan x \cdot \tan x} \cdot \sqrt{1 + \tan x \cdot \tan x}}}\right)} - 1 \]

      5. pow2 98.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1 - {\tan x}^{2}}{\color{blue}{{\left(\sqrt{1 + \tan x \cdot \tan x}\right)}^{2}}}\right)} - 1 \]

      6. hypot-1-def 98.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1 - {\tan x}^{2}}{{\color{blue}{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}}^{2}}\right)} - 1 \]

   3. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 - {\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}\right)} - 1} \]

   4. Taylor expanded in x around 0 21.3%

      \[\leadsto e^{\color{blue}{\log 2 + -1 \cdot {x}^{2}}} - 1 \]
   5. Step-by-step derivation

      1. metadata-eval 21.3%

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

      2. metadata-eval 21.3%

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

      3. *-inverses 21.3%

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

      4. log1p-def 21.3%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left({\left(\frac{\mathsf{hypot}\left(1, \tan x\right)}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2}\right)} + -1 \cdot {x}^{2}} - 1 \]

      5. mul-1-neg 21.3%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\mathsf{hypot}\left(1, \tan x\right)}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2}\right) + \color{blue}{\left(-{x}^{2}\right)}} - 1 \]

      6. unsub-neg 21.3%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left({\left(\frac{\mathsf{hypot}\left(1, \tan x\right)}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2}\right) - {x}^{2}}} - 1 \]

      7. log1p-def 21.3%

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

      8. *-inverses 21.3%

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

      9. metadata-eval 21.3%

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

      10. metadata-eval 21.3%

          \[\leadsto e^{\log \color{blue}{2} - {x}^{2}} - 1 \]

   6. Simplified 21.3%

      \[\leadsto e^{\color{blue}{\log 2 - {x}^{2}}} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1:\\ \;\;\;\;{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;e^{\log 2 - {x}^{2}} + -1\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.8× 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. add-log-exp 97.9%

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

   2. *-un-lft-identity 97.9%

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

   3. log-prod 97.9%

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

   4. metadata-eval 97.9%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

5. Applied egg-rr 99.5%

   \[\leadsto \frac{1 - \color{blue}{\left(0 + {\tan x}^{2}\right)}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
6. Step-by-step derivation

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. add-log-exp 97.9%

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

   2. *-un-lft-identity 97.9%

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

   3. log-prod 97.9%

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

   4. metadata-eval 97.9%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

5. Applied egg-rr 99.5%

   \[\leadsto \frac{1 - \color{blue}{\left(0 + {\tan x}^{2}\right)}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
6. Step-by-step derivation

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

   1. fma-udef 99.4%

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

   2. pow2 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 5: 55.2% 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 99.4%

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

   1. sub-neg 99.4%

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

   2. +-commutative 99.4%

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

   3. distribute-rgt-neg-in 99.4%

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

   4. fma-def 99.5%

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

3. Applied egg-rr 99.5%

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

   1. add-log-exp 97.9%

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

   2. *-un-lft-identity 97.9%

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

   3. log-prod 97.9%

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

   4. metadata-eval 97.9%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

5. Applied egg-rr 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \color{blue}{\left(0 + {\tan x}^{2}\right)}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0 54.4%

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

9. Final simplification 54.4%

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

Alternative 6: 58.5% 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 99.4%

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

   1. sub-neg 99.4%

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

   2. +-commutative 99.4%

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

   3. distribute-rgt-neg-in 99.4%

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

   4. fma-def 99.5%

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

3. Applied egg-rr 99.5%

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

   1. add-log-exp 97.9%

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

   2. *-un-lft-identity 97.9%

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

   3. log-prod 97.9%

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

   4. metadata-eval 97.9%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

5. Applied egg-rr 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \color{blue}{\left(0 + {\tan x}^{2}\right)}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0 54.4%

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

   1. +-commutative 54.4%

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

   2. unpow2 54.4%

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

   3. fma-def 54.4%

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

   4. add-sqr-sqrt 28.8%

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

   5. sqrt-prod 55.5%

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

   6. sqr-neg 55.5%

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

   7. sqrt-unprod 26.7%

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

   8. add-sqr-sqrt 57.3%

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

   9. fma-udef 57.3%

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

   10. distribute-rgt-neg-in 57.3%

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

   11. unpow2 57.3%

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

   12. +-commutative 57.3%

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

   13. sub-neg 57.3%

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

10. Applied egg-rr 57.3%

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

11. Final simplification 57.3%

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

Alternative 7: 54.8% accurate, 411.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.4%

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

   1. sub-neg 99.4%

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

   2. +-commutative 99.4%

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

   3. distribute-rgt-neg-in 99.4%

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

   4. fma-def 99.5%

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

3. Applied egg-rr 99.5%

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

   1. add-log-exp 97.9%

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

   2. *-un-lft-identity 97.9%

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

   3. log-prod 97.9%

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

   4. metadata-eval 97.9%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

5. Applied egg-rr 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \color{blue}{\left(0 + {\tan x}^{2}\right)}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0 54.0%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 54.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023334 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))