tan-example (used to crash)

Percentage Accurate: 79.0% → 99.7%

Time: 25.5s

Alternatives: 8

Speedup: 1.0×

Specification

\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]

Math

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (-
   (/ (+ (tan y) (tan z)) (+ 1.0 (- -1.0 (fma (tan y) (tan z) -1.0))))
   (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 + (-1.0 - fma(tan(y), tan(z), -1.0)))) - tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 + Float64(-1.0 - fma(tan(y), tan(z), -1.0)))) - tan(a)))
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

3. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

5. Simplified 99.7%

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

   1. expm1-log1p-u 95.0%

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

   2. expm1-udef 95.0%

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

   3. log1p-udef 95.0%

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

   4. add-exp-log 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate--l+ 99.7%

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

   2. fma-neg 99.7%

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

   3. metadata-eval 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

    \[\leadsto x + \left(\frac{\tan y + \tan z}{1 + \left(-1 - \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)} - \tan a\right) \]

Alternative 2: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-21}:\\ \;\;\;\;\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sin \left(y + z\right) \cdot \frac{1}{\cos \left(y + z\right)} - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -5e-11)
   (+ x (- (tan (+ y z)) (tan a)))
   (if (<= (tan a) 1e-21)
     (+ (* (+ (tan y) (tan z)) (/ 1.0 (- 1.0 (* (tan y) (tan z))))) (- x a))
     (+ x (- (* (sin (+ y z)) (/ 1.0 (cos (+ y z)))) (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -5e-11) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (tan(a) <= 1e-21) {
		tmp = ((tan(y) + tan(z)) * (1.0 / (1.0 - (tan(y) * tan(z))))) + (x - a);
	} else {
		tmp = x + ((sin((y + z)) * (1.0 / cos((y + z)))) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (tan(a) <= (-5d-11)) then
        tmp = x + (tan((y + z)) - tan(a))
    else if (tan(a) <= 1d-21) then
        tmp = ((tan(y) + tan(z)) * (1.0d0 / (1.0d0 - (tan(y) * tan(z))))) + (x - a)
    else
        tmp = x + ((sin((y + z)) * (1.0d0 / cos((y + z)))) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (Math.tan(a) <= -5e-11) {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	} else if (Math.tan(a) <= 1e-21) {
		tmp = ((Math.tan(y) + Math.tan(z)) * (1.0 / (1.0 - (Math.tan(y) * Math.tan(z))))) + (x - a);
	} else {
		tmp = x + ((Math.sin((y + z)) * (1.0 / Math.cos((y + z)))) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -5e-11:
		tmp = x + (math.tan((y + z)) - math.tan(a))
	elif math.tan(a) <= 1e-21:
		tmp = ((math.tan(y) + math.tan(z)) * (1.0 / (1.0 - (math.tan(y) * math.tan(z))))) + (x - a)
	else:
		tmp = x + ((math.sin((y + z)) * (1.0 / math.cos((y + z)))) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -5e-11)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (tan(a) <= 1e-21)
		tmp = Float64(Float64(Float64(tan(y) + tan(z)) * Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z))))) + Float64(x - a));
	else
		tmp = Float64(x + Float64(Float64(sin(Float64(y + z)) * Float64(1.0 / cos(Float64(y + z)))) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -5e-11)
		tmp = x + (tan((y + z)) - tan(a));
	elseif (tan(a) <= 1e-21)
		tmp = ((tan(y) + tan(z)) * (1.0 / (1.0 - (tan(y) * tan(z))))) + (x - a);
	else
		tmp = x + ((sin((y + z)) * (1.0 / cos((y + z)))) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -5e-11], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-21], N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -5.00000000000000018e-11

   1. Initial program 86.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

   if -5.00000000000000018e-11 < (tan.f64 a) < 9.99999999999999908e-22

   1. Initial program 80.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. associate-+r- 80.5%

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      2. +-commutative 80.5%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) + x\right)} - \tan a \]

      3. associate--l+ 80.5%

         \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(x - \tan a\right)} \]

   3. Simplified 80.5%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(x - \tan a\right)} \]

   4. Taylor expanded in a around 0 80.5%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(x + -1 \cdot a\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 80.5%

         \[\leadsto \tan \left(y + z\right) + \left(x + \color{blue}{\left(-a\right)}\right) \]

      2. unsub-neg 80.5%

         \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(x - a\right)} \]

   6. Simplified 80.5%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(x - a\right)} \]
   7. Step-by-step derivation

      1. tan-sum 99.8%

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

      2. div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

   if 9.99999999999999908e-22 < (tan.f64 a)

   1. Initial program 83.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. tan-quot 83.4%

         \[\leadsto \color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} + \left(x - \tan a\right) \]

      2. div-inv 83.4%

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

   3. Applied egg-rr 83.5%

      \[\leadsto x + \left(\color{blue}{\sin \left(y + z\right) \cdot \frac{1}{\cos \left(y + z\right)}} - \tan a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-21}:\\ \;\;\;\;\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sin \left(y + z\right) \cdot \frac{1}{\cos \left(y + z\right)} - \tan a\right)\\ \end{array} \]

Alternative 3: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-21}:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sin \left(y + z\right) \cdot \frac{1}{\cos \left(y + z\right)} - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -5e-11)
   (+ x (- (tan (+ y z)) (tan a)))
   (if (<= (tan a) 1e-21)
     (+ (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (- x a))
     (+ x (- (* (sin (+ y z)) (/ 1.0 (cos (+ y z)))) (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -5e-11) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (tan(a) <= 1e-21) {
		tmp = ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) + (x - a);
	} else {
		tmp = x + ((sin((y + z)) * (1.0 / cos((y + z)))) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (tan(a) <= (-5d-11)) then
        tmp = x + (tan((y + z)) - tan(a))
    else if (tan(a) <= 1d-21) then
        tmp = ((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) + (x - a)
    else
        tmp = x + ((sin((y + z)) * (1.0d0 / cos((y + z)))) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (Math.tan(a) <= -5e-11) {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	} else if (Math.tan(a) <= 1e-21) {
		tmp = ((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) + (x - a);
	} else {
		tmp = x + ((Math.sin((y + z)) * (1.0 / Math.cos((y + z)))) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -5e-11:
		tmp = x + (math.tan((y + z)) - math.tan(a))
	elif math.tan(a) <= 1e-21:
		tmp = ((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) + (x - a)
	else:
		tmp = x + ((math.sin((y + z)) * (1.0 / math.cos((y + z)))) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -5e-11)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (tan(a) <= 1e-21)
		tmp = Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) + Float64(x - a));
	else
		tmp = Float64(x + Float64(Float64(sin(Float64(y + z)) * Float64(1.0 / cos(Float64(y + z)))) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -5e-11)
		tmp = x + (tan((y + z)) - tan(a));
	elseif (tan(a) <= 1e-21)
		tmp = ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) + (x - a);
	else
		tmp = x + ((sin((y + z)) * (1.0 / cos((y + z)))) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -5e-11], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-21], N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -5.00000000000000018e-11

   1. Initial program 86.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

   if -5.00000000000000018e-11 < (tan.f64 a) < 9.99999999999999908e-22

   1. Initial program 80.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. associate-+r- 80.5%

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      2. +-commutative 80.5%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) + x\right)} - \tan a \]

      3. associate--l+ 80.5%

         \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(x - \tan a\right)} \]

   3. Simplified 80.5%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(x - \tan a\right)} \]

   4. Taylor expanded in a around 0 80.5%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(x + -1 \cdot a\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 80.5%

         \[\leadsto \tan \left(y + z\right) + \left(x + \color{blue}{\left(-a\right)}\right) \]

      2. unsub-neg 80.5%

         \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(x - a\right)} \]

   6. Simplified 80.5%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(x - a\right)} \]
   7. Step-by-step derivation

      1. tan-sum 99.8%

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

      2. div-inv 99.8%

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

      3. fma-def 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. associate-*r/ 99.8%

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

      3. *-rgt-identity 99.8%

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

   10. Simplified 99.8%

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

   if 9.99999999999999908e-22 < (tan.f64 a)

   1. Initial program 83.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. tan-quot 83.4%

         \[\leadsto \color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} + \left(x - \tan a\right) \]

      2. div-inv 83.4%

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

   3. Applied egg-rr 83.5%

      \[\leadsto x + \left(\color{blue}{\sin \left(y + z\right) \cdot \frac{1}{\cos \left(y + z\right)}} - \tan a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-21}:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sin \left(y + z\right) \cdot \frac{1}{\cos \left(y + z\right)} - \tan a\right)\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - tan(a));
}

Fortran

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

Java

public static double code(double x, double y, double z, double a) {
	return x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

3. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \]

Alternative 5: 59.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05 \lor \neg \left(\tan a \leq 0.02\right):\\ \;\;\;\;\sin z + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.05) (not (<= (tan a) 0.02)))
   (+ (sin z) (- x (tan a)))
   (+ (tan (+ y z)) (- x a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.05) || !(tan(a) <= 0.02)) {
		tmp = sin(z) + (x - tan(a));
	} else {
		tmp = tan((y + z)) + (x - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((tan(a) <= (-0.05d0)) .or. (.not. (tan(a) <= 0.02d0))) then
        tmp = sin(z) + (x - tan(a))
    else
        tmp = tan((y + z)) + (x - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((Math.tan(a) <= -0.05) || !(Math.tan(a) <= 0.02)) {
		tmp = Math.sin(z) + (x - Math.tan(a));
	} else {
		tmp = Math.tan((y + z)) + (x - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -0.05) or not (math.tan(a) <= 0.02):
		tmp = math.sin(z) + (x - math.tan(a))
	else:
		tmp = math.tan((y + z)) + (x - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -0.05) || !(tan(a) <= 0.02))
		tmp = Float64(sin(z) + Float64(x - tan(a)));
	else
		tmp = Float64(tan(Float64(y + z)) + Float64(x - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((tan(a) <= -0.05) || ~((tan(a) <= 0.02)))
		tmp = sin(z) + (x - tan(a));
	else
		tmp = tan((y + z)) + (x - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.05], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 0.02]], $MachinePrecision]], N[(N[Sin[z], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.050000000000000003 or 0.0200000000000000004 < (tan.f64 a)

   1. Initial program 83.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. associate-+r- 83.4%

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      2. +-commutative 83.4%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) + x\right)} - \tan a \]

      3. associate--l+ 83.4%

         \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(x - \tan a\right)} \]

   3. Simplified 83.4%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(x - \tan a\right)} \]
   4. Step-by-step derivation

      1. tan-quot 83.4%

         \[\leadsto \color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} + \left(x - \tan a\right) \]

      2. div-inv 83.4%

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

   5. Applied egg-rr 83.4%

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

   6. Taylor expanded in z around 0 64.4%

      \[\leadsto \sin \left(y + z\right) \cdot \frac{1}{\color{blue}{\cos y + -1 \cdot \left(z \cdot \sin y\right)}} + \left(x - \tan a\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 64.4%

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

      2. unsub-neg 64.4%

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

   8. Simplified 64.4%

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

   9. Taylor expanded in y around 0 42.4%

      \[\leadsto \color{blue}{\sin z} + \left(x - \tan a\right) \]

   if -0.050000000000000003 < (tan.f64 a) < 0.0200000000000000004

   1. Initial program 81.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. associate-+r- 81.8%

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      2. +-commutative 81.8%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) + x\right)} - \tan a \]

      3. associate--l+ 81.8%

         \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(x - \tan a\right)} \]

   3. Simplified 81.8%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(x - \tan a\right)} \]

   4. Taylor expanded in a around 0 81.2%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(x + -1 \cdot a\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 81.2%

         \[\leadsto \tan \left(y + z\right) + \left(x + \color{blue}{\left(-a\right)}\right) \]

      2. unsub-neg 81.2%

         \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(x - a\right)} \]

   6. Simplified 81.2%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(x - a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05 \lor \neg \left(\tan a \leq 0.02\right):\\ \;\;\;\;\sin z + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \]

Alternative 6: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

2. Final simplification 82.6%

   \[\leadsto x + \left(\tan \left(y + z\right) - \tan a\right) \]

Alternative 7: 50.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.25) x (if (<= a 1.55) (+ (tan (+ y z)) (- x a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.25) {
		tmp = x;
	} else if (a <= 1.55) {
		tmp = tan((y + z)) + (x - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-1.25d0)) then
        tmp = x
    else if (a <= 1.55d0) then
        tmp = tan((y + z)) + (x - a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.25) {
		tmp = x;
	} else if (a <= 1.55) {
		tmp = Math.tan((y + z)) + (x - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.25:
		tmp = x
	elif a <= 1.55:
		tmp = math.tan((y + z)) + (x - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.25)
		tmp = x;
	elseif (a <= 1.55)
		tmp = Float64(tan(Float64(y + z)) + Float64(x - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -1.25)
		tmp = x;
	elseif (a <= 1.55)
		tmp = tan((y + z)) + (x - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.25], x, If[LessEqual[a, 1.55], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.25 or 1.55000000000000004 < a

   1. Initial program 83.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

   2. Taylor expanded in x around inf 21.5%

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

   if -1.25 < a < 1.55000000000000004

   1. Initial program 81.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. associate-+r- 81.8%

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      2. +-commutative 81.8%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) + x\right)} - \tan a \]

      3. associate--l+ 81.8%

         \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(x - \tan a\right)} \]

   3. Simplified 81.8%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(x - \tan a\right)} \]

   4. Taylor expanded in a around 0 81.2%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(x + -1 \cdot a\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 81.2%

         \[\leadsto \tan \left(y + z\right) + \left(x + \color{blue}{\left(-a\right)}\right) \]

      2. unsub-neg 81.2%

         \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(x - a\right)} \]

   6. Simplified 81.2%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(x - a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 31.4% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 x)

C

double code(double x, double y, double z, double a) {
	return x;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x
end function

Java

public static double code(double x, double y, double z, double a) {
	return x;
}

Python

def code(x, y, z, a):
	return x

Julia

function code(x, y, z, a)
	return x
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x;
end

Wolfram

code[x_, y_, z_, a_] := x

Derivation

1. Initial program 82.6%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

2. Taylor expanded in x around inf 31.9%

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

3. Final simplification 31.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y z a)
  :name "tan-example (used to crash)"
  :precision binary64
  :pre (and (and (and (or (== x 0.0) (and (<= 0.5884142 x) (<= x 505.5909))) (or (and (<= -1.796658e+308 y) (<= y -9.425585e-310)) (and (<= 1.284938e-309 y) (<= y 1.751224e+308)))) (or (and (<= -1.776707e+308 z) (<= z -8.599796e-310)) (and (<= 3.293145e-311 z) (<= z 1.725154e+308)))) (or (and (<= -1.796658e+308 a) (<= a -9.425585e-310)) (and (<= 1.284938e-309 a) (<= a 1.751224e+308))))
  (+ x (- (tan (+ y z)) (tan a))))