Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A

Percentage Accurate: 88.0% → 99.5%

Time: 7.0s

Alternatives: 6

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Python

def code(x, y, z):
	return (x + y) / (1.0 - (y / z))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Python

def code(x, y, z):
	return (x + y) / (1.0 - (y / z))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-272} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-265}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-272) (not (<= t_0 5e-265)))
     t_0
     (* z (- -1.0 (/ x y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-272) || !(t_0 <= 5e-265)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-5d-272)) .or. (.not. (t_0 <= 5d-265))) then
        tmp = t_0
    else
        tmp = z * ((-1.0d0) - (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-272) || !(t_0 <= 5e-265)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -5e-272) or not (t_0 <= 5e-265):
		tmp = t_0
	else:
		tmp = z * (-1.0 - (x / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -5e-272) || !(t_0 <= 5e-265))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -5e-272) || ~((t_0 <= 5e-265)))
		tmp = t_0;
	else
		tmp = z * (-1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-272], N[Not[LessEqual[t$95$0, 5e-265]], $MachinePrecision]], t$95$0, N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -4.99999999999999982e-272 or 5.0000000000000001e-265 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   if -4.99999999999999982e-272 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 5.0000000000000001e-265

   1. Initial program 24.6%

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

      1. clear-num 24.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{1 - \frac{y}{z}}{x + y}}} \]

      2. associate-/r/ 24.7%

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

   4. Applied egg-rr 24.7%

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

      1. associate-*l/ 24.6%

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

      2. *-un-lft-identity 24.6%

         \[\leadsto \frac{\color{blue}{x + y}}{1 - \frac{y}{z}} \]

      3. clear-num 24.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{1 - \frac{y}{z}}{x + y}}} \]

      4. +-commutative 24.6%

         \[\leadsto \frac{1}{\frac{1 - \frac{y}{z}}{\color{blue}{y + x}}} \]

   6. Applied egg-rr 24.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 - \frac{y}{z}}{y + x}}} \]

   7. Taylor expanded in z around 0 81.6%

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

      1. mul-1-neg 81.6%

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

      2. associate-/l* 99.8%

         \[\leadsto -\color{blue}{z \cdot \frac{x + y}{y}} \]

      3. +-commutative 99.8%

         \[\leadsto -z \cdot \frac{\color{blue}{y + x}}{y} \]

      4. distribute-rgt-neg-in 99.8%

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

      5. neg-sub0 99.8%

         \[\leadsto z \cdot \color{blue}{\left(0 - \frac{y + x}{y}\right)} \]

      6. *-lft-identity 99.8%

         \[\leadsto z \cdot \left(0 - \frac{\color{blue}{1 \cdot \left(y + x\right)}}{y}\right) \]

      7. associate-*l/ 99.9%

         \[\leadsto z \cdot \left(0 - \color{blue}{\frac{1}{y} \cdot \left(y + x\right)}\right) \]

      8. distribute-lft-in 99.9%

         \[\leadsto z \cdot \left(0 - \color{blue}{\left(\frac{1}{y} \cdot y + \frac{1}{y} \cdot x\right)}\right) \]

      9. lft-mult-inverse 99.9%

         \[\leadsto z \cdot \left(0 - \left(\color{blue}{1} + \frac{1}{y} \cdot x\right)\right) \]

      10. lft-mult-inverse 99.6%

          \[\leadsto z \cdot \left(0 - \left(\color{blue}{\frac{1}{x} \cdot x} + \frac{1}{y} \cdot x\right)\right) \]

      11. distribute-rgt-in 99.6%

          \[\leadsto z \cdot \left(0 - \color{blue}{x \cdot \left(\frac{1}{x} + \frac{1}{y}\right)}\right) \]

      12. distribute-lft-in 99.6%

          \[\leadsto z \cdot \left(0 - \color{blue}{\left(x \cdot \frac{1}{x} + x \cdot \frac{1}{y}\right)}\right) \]

      13. rgt-mult-inverse 99.9%

          \[\leadsto z \cdot \left(0 - \left(\color{blue}{1} + x \cdot \frac{1}{y}\right)\right) \]

      14. associate-*r/ 99.9%

          \[\leadsto z \cdot \left(0 - \left(1 + \color{blue}{\frac{x \cdot 1}{y}}\right)\right) \]

      15. *-rgt-identity 99.9%

          \[\leadsto z \cdot \left(0 - \left(1 + \frac{\color{blue}{x}}{y}\right)\right) \]

      16. associate--r+ 99.9%

          \[\leadsto z \cdot \color{blue}{\left(\left(0 - 1\right) - \frac{x}{y}\right)} \]

      17. metadata-eval 99.9%

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

   9. Simplified 99.9%

      \[\leadsto \color{blue}{z \cdot \left(-1 - \frac{x}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-272} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 5 \cdot 10^{-265}\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.04 \cdot 10^{+65}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-27}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+112}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.04e+65)
   (- z)
   (if (<= y -2e-27) y (if (<= y 2.75e-96) x (if (<= y 1.06e+112) y (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.04e+65) {
		tmp = -z;
	} else if (y <= -2e-27) {
		tmp = y;
	} else if (y <= 2.75e-96) {
		tmp = x;
	} else if (y <= 1.06e+112) {
		tmp = y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.04d+65)) then
        tmp = -z
    else if (y <= (-2d-27)) then
        tmp = y
    else if (y <= 2.75d-96) then
        tmp = x
    else if (y <= 1.06d+112) then
        tmp = y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.04e+65) {
		tmp = -z;
	} else if (y <= -2e-27) {
		tmp = y;
	} else if (y <= 2.75e-96) {
		tmp = x;
	} else if (y <= 1.06e+112) {
		tmp = y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.04e+65:
		tmp = -z
	elif y <= -2e-27:
		tmp = y
	elif y <= 2.75e-96:
		tmp = x
	elif y <= 1.06e+112:
		tmp = y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.04e+65)
		tmp = Float64(-z);
	elseif (y <= -2e-27)
		tmp = y;
	elseif (y <= 2.75e-96)
		tmp = x;
	elseif (y <= 1.06e+112)
		tmp = y;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.04e+65)
		tmp = -z;
	elseif (y <= -2e-27)
		tmp = y;
	elseif (y <= 2.75e-96)
		tmp = x;
	elseif (y <= 1.06e+112)
		tmp = y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.04e+65], (-z), If[LessEqual[y, -2e-27], y, If[LessEqual[y, 2.75e-96], x, If[LessEqual[y, 1.06e+112], y, (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.03999999999999999e65 or 1.06e112 < y

   1. Initial program 73.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 73.5%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. neg-mul-1 73.5%

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 73.5%

      \[\leadsto \color{blue}{-z} \]

   if -1.03999999999999999e65 < y < -2.0000000000000001e-27 or 2.7499999999999998e-96 < y < 1.06e112

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 57.8%

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative 57.8%

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

   5. Simplified 57.8%

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

   6. Taylor expanded in y around inf 44.9%

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

   if -2.0000000000000001e-27 < y < 2.7499999999999998e-96

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 65.2%

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

Alternative 3: 72.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -145000000000 \lor \neg \left(z \leq 3.5 \cdot 10^{-24}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -145000000000.0) (not (<= z 3.5e-24)))
   (+ x y)
   (* z (- -1.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -145000000000.0) || !(z <= 3.5e-24)) {
		tmp = x + y;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-145000000000.0d0)) .or. (.not. (z <= 3.5d-24))) then
        tmp = x + y
    else
        tmp = z * ((-1.0d0) - (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -145000000000.0) || !(z <= 3.5e-24)) {
		tmp = x + y;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -145000000000.0) or not (z <= 3.5e-24):
		tmp = x + y
	else:
		tmp = z * (-1.0 - (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -145000000000.0) || !(z <= 3.5e-24))
		tmp = Float64(x + y);
	else
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -145000000000.0) || ~((z <= 3.5e-24)))
		tmp = x + y;
	else
		tmp = z * (-1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -145000000000.0], N[Not[LessEqual[z, 3.5e-24]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e11 or 3.4999999999999996e-24 < z

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 83.0%

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative 83.0%

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

   5. Simplified 83.0%

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

   if -1.45e11 < z < 3.4999999999999996e-24

   1. Initial program 84.7%

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

      1. clear-num 84.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{1 - \frac{y}{z}}{x + y}}} \]

      2. associate-/r/ 84.6%

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

   4. Applied egg-rr 84.6%

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

      1. associate-*l/ 84.7%

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

      2. *-un-lft-identity 84.7%

         \[\leadsto \frac{\color{blue}{x + y}}{1 - \frac{y}{z}} \]

      3. clear-num 84.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{1 - \frac{y}{z}}{x + y}}} \]

      4. +-commutative 84.4%

         \[\leadsto \frac{1}{\frac{1 - \frac{y}{z}}{\color{blue}{y + x}}} \]

   6. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 - \frac{y}{z}}{y + x}}} \]

   7. Taylor expanded in z around 0 67.1%

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

      1. mul-1-neg 67.1%

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

      2. associate-/l* 75.3%

         \[\leadsto -\color{blue}{z \cdot \frac{x + y}{y}} \]

      3. +-commutative 75.3%

         \[\leadsto -z \cdot \frac{\color{blue}{y + x}}{y} \]

      4. distribute-rgt-neg-in 75.3%

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

      5. neg-sub0 75.3%

         \[\leadsto z \cdot \color{blue}{\left(0 - \frac{y + x}{y}\right)} \]

      6. *-lft-identity 75.3%

         \[\leadsto z \cdot \left(0 - \frac{\color{blue}{1 \cdot \left(y + x\right)}}{y}\right) \]

      7. associate-*l/ 75.2%

         \[\leadsto z \cdot \left(0 - \color{blue}{\frac{1}{y} \cdot \left(y + x\right)}\right) \]

      8. distribute-lft-in 75.2%

         \[\leadsto z \cdot \left(0 - \color{blue}{\left(\frac{1}{y} \cdot y + \frac{1}{y} \cdot x\right)}\right) \]

      9. lft-mult-inverse 75.3%

         \[\leadsto z \cdot \left(0 - \left(\color{blue}{1} + \frac{1}{y} \cdot x\right)\right) \]

      10. lft-mult-inverse 75.2%

          \[\leadsto z \cdot \left(0 - \left(\color{blue}{\frac{1}{x} \cdot x} + \frac{1}{y} \cdot x\right)\right) \]

      11. distribute-rgt-in 75.2%

          \[\leadsto z \cdot \left(0 - \color{blue}{x \cdot \left(\frac{1}{x} + \frac{1}{y}\right)}\right) \]

      12. distribute-lft-in 75.2%

          \[\leadsto z \cdot \left(0 - \color{blue}{\left(x \cdot \frac{1}{x} + x \cdot \frac{1}{y}\right)}\right) \]

      13. rgt-mult-inverse 75.3%

          \[\leadsto z \cdot \left(0 - \left(\color{blue}{1} + x \cdot \frac{1}{y}\right)\right) \]

      14. associate-*r/ 75.4%

          \[\leadsto z \cdot \left(0 - \left(1 + \color{blue}{\frac{x \cdot 1}{y}}\right)\right) \]

      15. *-rgt-identity 75.4%

          \[\leadsto z \cdot \left(0 - \left(1 + \frac{\color{blue}{x}}{y}\right)\right) \]

      16. associate--r+ 75.4%

          \[\leadsto z \cdot \color{blue}{\left(\left(0 - 1\right) - \frac{x}{y}\right)} \]

      17. metadata-eval 75.4%

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

   9. Simplified 75.4%

      \[\leadsto \color{blue}{z \cdot \left(-1 - \frac{x}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -145000000000 \lor \neg \left(z \leq 3.5 \cdot 10^{-24}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+108} \lor \neg \left(y \leq 1.35 \cdot 10^{+112}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1e+108) (not (<= y 1.35e+112))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e+108) || !(y <= 1.35e+112)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1d+108)) .or. (.not. (y <= 1.35d+112))) then
        tmp = -z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e+108) || !(y <= 1.35e+112)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1e+108) or not (y <= 1.35e+112):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1e+108) || !(y <= 1.35e+112))
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1e+108) || ~((y <= 1.35e+112)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1e+108], N[Not[LessEqual[y, 1.35e+112]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e108 or 1.3500000000000001e112 < y

   1. Initial program 70.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 77.0%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. neg-mul-1 77.0%

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{-z} \]

   if -1e108 < y < 1.3500000000000001e112

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 69.2%

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative 69.2%

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

   5. Simplified 69.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+108} \lor \neg \left(y \leq 1.35 \cdot 10^{+112}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-137}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-178}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.5e-137) x (if (<= x 2.35e-178) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.5e-137) {
		tmp = x;
	} else if (x <= 2.35e-178) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-8.5d-137)) then
        tmp = x
    else if (x <= 2.35d-178) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.5e-137) {
		tmp = x;
	} else if (x <= 2.35e-178) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8.5e-137:
		tmp = x
	elif x <= 2.35e-178:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.5e-137)
		tmp = x;
	elseif (x <= 2.35e-178)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8.5e-137)
		tmp = x;
	elseif (x <= 2.35e-178)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.5e-137], x, If[LessEqual[x, 2.35e-178], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5000000000000001e-137 or 2.35e-178 < x

   1. Initial program 91.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 47.8%

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

   if -8.5000000000000001e-137 < x < 2.35e-178

   1. Initial program 94.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 58.9%

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative 58.9%

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

   5. Simplified 58.9%

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

   6. Taylor expanded in y around inf 48.0%

      \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 34.8% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 92.6%

   \[\frac{x + y}{1 - \frac{y}{z}} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 38.2%

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

Developer Target 1: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{-y} \cdot z\\ \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (+ y x) (- y)) z)))
   (if (< y -3.7429310762689856e+171)
     t_0
     (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y + x) / -y) * z
    if (y < (-3.7429310762689856d+171)) then
        tmp = t_0
    else if (y < 3.5534662456086734d+168) then
        tmp = (x + y) / (1.0d0 - (y / z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y + x) / -y) * z
	tmp = 0
	if y < -3.7429310762689856e+171:
		tmp = t_0
	elif y < 3.5534662456086734e+168:
		tmp = (x + y) / (1.0 - (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y + x) / Float64(-y)) * z)
	tmp = 0.0
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y + x) / -y) * z;
	tmp = 0.0;
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = (x + y) / (1.0 - (y / z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] / (-y)), $MachinePrecision] * z), $MachinePrecision]}, If[Less[y, -3.7429310762689856e+171], t$95$0, If[Less[y, 3.5534662456086734e+168], N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024165 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -3742931076268985600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (/ (+ y x) (- y)) z) (if (< y 3553466245608673400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z))))

  (/ (+ x y) (- 1.0 (/ y z))))