tan-example (used to crash)

Percentage Accurate: 79.0% → 99.6%

Time: 45.8s

Alternatives: 9

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: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y \cdot \tan z\\ t_1 := \frac{\cos a}{\sin a}\\ x + \frac{\left(\tan y + \tan z\right) \cdot t\_1 + \left(t\_0 + -1\right)}{t\_1 \cdot \left(1 - t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* (tan y) (tan z))) (t_1 (/ (cos a) (sin a))))
   (+ x (/ (+ (* (+ (tan y) (tan z)) t_1) (+ t_0 -1.0)) (* t_1 (- 1.0 t_0))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) * tan(z);
	double t_1 = cos(a) / sin(a);
	return x + ((((tan(y) + tan(z)) * t_1) + (t_0 + -1.0)) / (t_1 * (1.0 - t_0)));
}

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) :: t_0
    real(8) :: t_1
    t_0 = tan(y) * tan(z)
    t_1 = cos(a) / sin(a)
    code = x + ((((tan(y) + tan(z)) * t_1) + (t_0 + (-1.0d0))) / (t_1 * (1.0d0 - t_0)))
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) * Math.tan(z);
	double t_1 = Math.cos(a) / Math.sin(a);
	return x + ((((Math.tan(y) + Math.tan(z)) * t_1) + (t_0 + -1.0)) / (t_1 * (1.0 - t_0)));
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) * math.tan(z)
	t_1 = math.cos(a) / math.sin(a)
	return x + ((((math.tan(y) + math.tan(z)) * t_1) + (t_0 + -1.0)) / (t_1 * (1.0 - t_0)))

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) * tan(z))
	t_1 = Float64(cos(a) / sin(a))
	return Float64(x + Float64(Float64(Float64(Float64(tan(y) + tan(z)) * t_1) + Float64(t_0 + -1.0)) / Float64(t_1 * Float64(1.0 - t_0))))
end

MATLAB

function tmp = code(x, y, z, a)
	t_0 = tan(y) * tan(z);
	t_1 = cos(a) / sin(a);
	tmp = x + ((((tan(y) + tan(z)) * t_1) + (t_0 + -1.0)) / (t_1 * (1.0 - t_0)));
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[a], $MachinePrecision] / N[Sin[a], $MachinePrecision]), $MachinePrecision]}, N[(x + N[(N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 77.0%

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

   1. add-sqr-sqrt 36.2%

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

   2. sqrt-unprod 56.8%

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

   3. pow2 56.8%

      \[\leadsto x + \left(\sqrt{\color{blue}{{\tan \left(y + z\right)}^{2}}} - \tan a\right) \]

4. Applied egg-rr 56.8%

   \[\leadsto x + \left(\color{blue}{\sqrt{{\tan \left(y + z\right)}^{2}}} - \tan a\right) \]
5. Step-by-step derivation

   1. sqrt-pow1 77.0%

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

   2. metadata-eval 77.0%

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

   3. pow1 77.0%

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

   4. tan-sum 99.7%

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

   5. tan-quot 99.7%

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

   6. clear-num 99.7%

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

   7. frac-sub 99.7%

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

6. Applied egg-rr 99.7%

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

   1. *-rgt-identity 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

   \[\leadsto x + \frac{\left(\tan y + \tan z\right) \cdot \frac{\cos a}{\sin a} + \left(\tan y \cdot \tan z + -1\right)}{\frac{\cos a}{\sin a} \cdot \left(1 - \tan y \cdot \tan z\right)} \]
10. Add Preprocessing

Alternative 2: 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 77.0%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. 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%

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

4. Applied egg-rr 99.7%

   \[\leadsto x + \left(\color{blue}{\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
5. 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%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -9.2e-8) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -9.2e-8) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(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 (y <= (-9.2d-8)) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(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 (y <= -9.2e-8) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -9.2e-8:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -9.2e-8)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (y <= -9.2e-8)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -9.2e-8], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.2000000000000003e-8

   1. Initial program 59.5%

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

      1. add-exp-log 53.5%

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

      2. +-commutative 53.5%

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

      3. associate-+l- 53.5%

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

   4. Applied egg-rr 53.5%

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

   5. Taylor expanded in z around 0 53.4%

      \[\leadsto e^{\log \left(\color{blue}{\frac{\sin y}{\cos y}} - \left(\tan a - x\right)\right)} \]
   6. Step-by-step derivation

      1. rem-exp-log 59.5%

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

      2. tan-quot 59.4%

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

      3. associate--r- 59.5%

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

   7. Applied egg-rr 59.5%

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

   if -9.2000000000000003e-8 < y

   1. Initial program 83.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 69.7%

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

      1. tan-quot 69.7%

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

      2. *-un-lft-identity 69.7%

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

   5. Applied egg-rr 69.7%

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

      1. *-lft-identity 69.7%

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

   7. Simplified 69.7%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 35.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.6:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 1.6) (+ x (- z (tan a))) (pow (cbrt x) 3.0)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.6) {
		tmp = x + (z - tan(a));
	} else {
		tmp = pow(cbrt(x), 3.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.6) {
		tmp = x + (z - Math.tan(a));
	} else {
		tmp = Math.pow(Math.cbrt(x), 3.0);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 1.6)
		tmp = Float64(x + Float64(z - tan(a)));
	else
		tmp = cbrt(x) ^ 3.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 1.6], N[(x + N[(z - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[x, 1/3], $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.6000000000000001

   1. Initial program 82.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 55.7%

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

   4. Taylor expanded in z around 0 39.8%

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

   if 1.6000000000000001 < z

   1. Initial program 59.3%

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

      1. add-exp-log 55.4%

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

      2. +-commutative 55.4%

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

      3. associate-+l- 55.3%

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

   4. Applied egg-rr 55.3%

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

   5. Taylor expanded in x around inf 22.9%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 22.9%

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

      2. log-rec 22.9%

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

      3. remove-double-neg 22.9%

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

   7. Simplified 22.9%

      \[\leadsto e^{\color{blue}{\log x}} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 22.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{e^{\log x}} \cdot \sqrt[3]{e^{\log x}}\right) \cdot \sqrt[3]{e^{\log x}}} \]

      2. pow3 22.9%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{\log x}}\right)}^{3}} \]

      3. rem-exp-log 22.9%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x}}\right)}^{3} \]

   9. Applied egg-rr 22.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 77.0%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Final simplification 77.0%

   \[\leadsto x + \left(\tan \left(y + z\right) - \tan a\right) \]
4. Add Preprocessing

Alternative 6: 35.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.6:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 1.6) (+ x (- z (tan a))) (exp (log x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.6) {
		tmp = x + (z - tan(a));
	} else {
		tmp = exp(log(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 (z <= 1.6d0) then
        tmp = x + (z - tan(a))
    else
        tmp = exp(log(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.6) {
		tmp = x + (z - Math.tan(a));
	} else {
		tmp = Math.exp(Math.log(x));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 1.6:
		tmp = x + (z - math.tan(a))
	else:
		tmp = math.exp(math.log(x))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 1.6)
		tmp = Float64(x + Float64(z - tan(a)));
	else
		tmp = exp(log(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 1.6)
		tmp = x + (z - tan(a));
	else
		tmp = exp(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 1.6], N[(x + N[(z - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[x], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.6000000000000001

   1. Initial program 82.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 55.7%

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

   4. Taylor expanded in z around 0 39.8%

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

   if 1.6000000000000001 < z

   1. Initial program 59.3%

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

      1. add-exp-log 55.4%

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

      2. +-commutative 55.4%

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

      3. associate-+l- 55.3%

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

   4. Applied egg-rr 55.3%

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

   5. Taylor expanded in x around inf 22.9%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 22.9%

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

      2. log-rec 22.9%

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

      3. remove-double-neg 22.9%

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

   7. Simplified 22.9%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0 56.5%

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

   1. tan-quot 56.5%

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

   2. *-un-lft-identity 56.5%

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

5. Applied egg-rr 56.5%

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

   1. *-lft-identity 56.5%

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

7. Simplified 56.5%

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

8. Final simplification 56.5%

   \[\leadsto x + \left(\tan z - \tan a\right) \]
9. Add Preprocessing

Alternative 8: 35.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.8:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (if (<= z 1.8) (+ x (- z (tan a))) x))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.8) {
		tmp = x + (z - tan(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 (z <= 1.8d0) then
        tmp = x + (z - tan(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 (z <= 1.8) {
		tmp = x + (z - Math.tan(a));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 1.8:
		tmp = x + (z - math.tan(a))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 1.8)
		tmp = Float64(x + Float64(z - tan(a)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 1.8)
		tmp = x + (z - tan(a));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 1.8], N[(x + N[(z - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < 1.80000000000000004

   1. Initial program 82.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 55.7%

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

   4. Taylor expanded in z around 0 39.8%

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

   if 1.80000000000000004 < z

   1. Initial program 59.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 22.9%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.8:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 77.0%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 32.1%

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

4. Final simplification 32.1%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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))))