Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 8.7s

Alternatives: 4

Speedup: N/A×

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. add-log-exp 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. log-prod 99.3%

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

   4. metadata-eval 99.3%

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

   5. add-log-exp 99.5%

      \[\leadsto 0 + \color{blue}{\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}} \]

   6. pow2 99.5%

      \[\leadsto 0 + \frac{1 - \color{blue}{{\tan x}^{2}}}{1 + \tan x \cdot \tan x} \]

   7. pow2 99.5%

      \[\leadsto 0 + \frac{1 - {\tan x}^{2}}{1 + \color{blue}{{\tan x}^{2}}} \]

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{0 + \frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}}} \]
5. Step-by-step derivation

   1. +-lft-identity 99.5%

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

6. Simplified 99.5%

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

Alternative 2: 60.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;-1 + e^{\log 2 - {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(1\right)\right)}{1 + {\tan x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= (tan x) -1.0) (not (<= (tan x) 1.0)))
   (+ -1.0 (exp (- (log 2.0) (pow x 2.0))))
   (/ (log1p (expm1 1.0)) (+ 1.0 (pow (tan x) 2.0)))))

C

double code(double x) {
	double tmp;
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0)) {
		tmp = -1.0 + exp((log(2.0) - pow(x, 2.0)));
	} else {
		tmp = log1p(expm1(1.0)) / (1.0 + pow(tan(x), 2.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((Math.tan(x) <= -1.0) || !(Math.tan(x) <= 1.0)) {
		tmp = -1.0 + Math.exp((Math.log(2.0) - Math.pow(x, 2.0)));
	} else {
		tmp = Math.log1p(Math.expm1(1.0)) / (1.0 + Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.tan(x) <= -1.0) or not (math.tan(x) <= 1.0):
		tmp = -1.0 + math.exp((math.log(2.0) - math.pow(x, 2.0)))
	else:
		tmp = math.log1p(math.expm1(1.0)) / (1.0 + math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0))
		tmp = Float64(-1.0 + exp(Float64(log(2.0) - (x ^ 2.0))));
	else
		tmp = Float64(log1p(expm1(1.0)) / Float64(1.0 + (tan(x) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[N[Tan[x], $MachinePrecision], -1.0], N[Not[LessEqual[N[Tan[x], $MachinePrecision], 1.0]], $MachinePrecision]], N[(-1.0 + N[Exp[N[(N[Log[2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(Exp[1.0] - 1), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -1 or 1 < (tan.f64 x)

   1. Initial program 99.2%

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

      1. expm1-log1p-u 99.2%

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

      2. expm1-undefine 99.2%

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

      3. div-inv 99.0%

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

      4. div-inv 99.2%

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

      5. pow2 99.2%

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

      6. pow2 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in x around 0 21.0%

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

      1. mul-1-neg 21.0%

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

      2. unsub-neg 21.0%

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

   7. Simplified 21.0%

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

   if -1 < (tan.f64 x) < 1

   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. add-log-exp 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. log-prod 99.5%

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

      4. metadata-eval 99.5%

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

      5. add-log-exp 99.6%

         \[\leadsto 0 + \color{blue}{\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}} \]

      6. pow2 99.6%

         \[\leadsto 0 + \frac{1 - \color{blue}{{\tan x}^{2}}}{1 + \tan x \cdot \tan x} \]

      7. pow2 99.6%

         \[\leadsto 0 + \frac{1 - {\tan x}^{2}}{1 + \color{blue}{{\tan x}^{2}}} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{0 + \frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}}} \]
   5. Step-by-step derivation

      1. +-lft-identity 99.6%

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

   6. Simplified 99.6%

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

      1. log1p-expm1-u 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in x around 0 73.1%

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

       1. expm1-define 73.1%

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

   11. Simplified 73.1%

       \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(1\right)}\right)}{1 + {\tan x}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;-1 + e^{\log 2 - {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(1\right)\right)}{1 + {\tan x}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ -1 + e^{\log 2 - {x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ -1.0 (exp (- (log 2.0) (pow x 2.0)))))

C

double code(double x) {
	return -1.0 + exp((log(2.0) - pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) + exp((log(2.0d0) - (x ** 2.0d0)))
end function

Java

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

Python

def code(x):
	return -1.0 + math.exp((math.log(2.0) - math.pow(x, 2.0)))

Julia

function code(x)
	return Float64(-1.0 + exp(Float64(log(2.0) - (x ^ 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 + exp((log(2.0) - (x ^ 2.0)));
end

Wolfram

code[x_] := N[(-1.0 + N[Exp[N[(N[Log[2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $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. expm1-log1p-u 99.5%

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

   2. expm1-undefine 99.4%

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

   3. div-inv 99.3%

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

   4. div-inv 99.4%

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

   5. pow2 99.4%

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

   6. pow2 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0 53.6%

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

   1. mul-1-neg 53.6%

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

   2. unsub-neg 53.6%

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

7. Simplified 53.6%

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

8. Final simplification 53.6%

   \[\leadsto -1 + e^{\log 2 - {x}^{2}} \]
9. Add Preprocessing

Alternative 4: 55.2% 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 99.5%

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

3. Taylor expanded in x around 0 52.9%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024119 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))