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

Percentage Accurate: 87.8% → 99.5%

Time: 6.2s

Alternatives: 7

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: 87.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -2e-250)
     (/ (+ x y) (/ (- z y) z))
     (if (<= t_0 0.0) (* z (- -1.0 (/ x y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -2e-250) {
		tmp = (x + y) / ((z - y) / z);
	} else if (t_0 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} 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 = (x + y) / (1.0d0 - (y / z))
    if (t_0 <= (-2d-250)) then
        tmp = (x + y) / ((z - y) / z)
    else if (t_0 <= 0.0d0) then
        tmp = z * ((-1.0d0) - (x / y))
    else
        tmp = t_0
    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 <= -2e-250) {
		tmp = (x + y) / ((z - y) / z);
	} else if (t_0 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -2e-250:
		tmp = (x + y) / ((z - y) / z)
	elif t_0 <= 0.0:
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = t_0
	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 <= -2e-250)
		tmp = Float64(Float64(x + y) / Float64(Float64(z - y) / z));
	elseif (t_0 <= 0.0)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = t_0;
	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 <= -2e-250)
		tmp = (x + y) / ((z - y) / z);
	elseif (t_0 <= 0.0)
		tmp = z * (-1.0 - (x / y));
	else
		tmp = t_0;
	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[LessEqual[t$95$0, -2e-250], N[(N[(x + y), $MachinePrecision] / N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -2.0000000000000001e-250

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 99.8%

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

   if -2.0000000000000001e-250 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 10.6%

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

   3. Taylor expanded in z around 0 10.6%

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

   4. Taylor expanded in z around 0 96.6%

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

      1. mul-1-neg 96.6%

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

      2. associate-/l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. mul-1-neg 99.9%

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

      5. associate-*r/ 99.9%

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

      6. neg-mul-1 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. sub-neg 99.9%

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

      9. div-sub 99.9%

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

      10. distribute-neg-frac 99.9%

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

      11. neg-mul-1 99.9%

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

      12. *-inverses 99.9%

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

      13. sub-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. distribute-neg-in 99.9%

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

      16. +-commutative 99.9%

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

      17. distribute-neg-in 99.9%

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

      18. metadata-eval 99.9%

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

      19. unsub-neg 99.9%

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

   6. Simplified 99.9%

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

   if -0.0 < (/.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
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-250} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-250) (not (<= t_0 0.0))) 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 <= -2e-250) || !(t_0 <= 0.0)) {
		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 <= (-2d-250)) .or. (.not. (t_0 <= 0.0d0))) 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 <= -2e-250) || !(t_0 <= 0.0)) {
		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 <= -2e-250) or not (t_0 <= 0.0):
		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 <= -2e-250) || !(t_0 <= 0.0))
		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 <= -2e-250) || ~((t_0 <= 0.0)))
		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, -2e-250], N[Not[LessEqual[t$95$0, 0.0]], $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))) < -2.0000000000000001e-250 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.8%

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

   if -2.0000000000000001e-250 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 10.6%

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

   3. Taylor expanded in z around 0 10.6%

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

   4. Taylor expanded in z around 0 96.6%

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

      1. mul-1-neg 96.6%

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

      2. associate-/l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. mul-1-neg 99.9%

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

      5. associate-*r/ 99.9%

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

      6. neg-mul-1 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. sub-neg 99.9%

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

      9. div-sub 99.9%

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

      10. distribute-neg-frac 99.9%

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

      11. neg-mul-1 99.9%

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

      12. *-inverses 99.9%

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

      13. sub-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. distribute-neg-in 99.9%

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

      16. +-commutative 99.9%

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

      17. distribute-neg-in 99.9%

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

      18. metadata-eval 99.9%

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

      19. unsub-neg 99.9%

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

   6. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-250} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-27} \lor \neg \left(y \leq 1.85 \cdot 10^{-47}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5e-27) (not (<= y 1.85e-47)))
   (* z (- -1.0 (/ x y)))
   (/ x (- 1.0 (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e-27) || !(y <= 1.85e-47)) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = x / (1.0 - (y / 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 <= (-5d-27)) .or. (.not. (y <= 1.85d-47))) then
        tmp = z * ((-1.0d0) - (x / y))
    else
        tmp = x / (1.0d0 - (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e-27) || !(y <= 1.85e-47)) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = x / (1.0 - (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5e-27) or not (y <= 1.85e-47):
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = x / (1.0 - (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5e-27) || !(y <= 1.85e-47))
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5e-27) || ~((y <= 1.85e-47)))
		tmp = z * (-1.0 - (x / y));
	else
		tmp = x / (1.0 - (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5e-27], N[Not[LessEqual[y, 1.85e-47]], $MachinePrecision]], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.0000000000000002e-27 or 1.85e-47 < y

   1. Initial program 75.0%

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

   3. Taylor expanded in z around 0 75.0%

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

   4. Taylor expanded in z around 0 69.0%

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

      1. mul-1-neg 69.0%

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

      2. associate-/l* 77.4%

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

      3. distribute-rgt-neg-in 77.4%

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

      4. mul-1-neg 77.4%

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

      5. associate-*r/ 77.4%

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

      6. neg-mul-1 77.4%

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

      7. distribute-neg-in 77.4%

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

      8. sub-neg 77.4%

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

      9. div-sub 77.4%

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

      10. distribute-neg-frac 77.4%

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

      11. neg-mul-1 77.4%

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

      12. *-inverses 77.4%

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

      13. sub-neg 77.4%

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

      14. neg-mul-1 77.4%

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

      15. distribute-neg-in 77.4%

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

      16. +-commutative 77.4%

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

      17. distribute-neg-in 77.4%

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

      18. metadata-eval 77.4%

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

      19. unsub-neg 77.4%

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

   6. Simplified 77.4%

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

   if -5.0000000000000002e-27 < y < 1.85e-47

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 88.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-27} \lor \neg \left(y \leq 1.85 \cdot 10^{-47}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-27} \lor \neg \left(y \leq 1.02 \cdot 10^{-38}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.6e-27) (not (<= y 1.02e-38)))
   (* z (- -1.0 (/ x y)))
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.6e-27) || !(y <= 1.02e-38)) {
		tmp = z * (-1.0 - (x / y));
	} 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 <= (-2.6d-27)) .or. (.not. (y <= 1.02d-38))) then
        tmp = z * ((-1.0d0) - (x / y))
    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 <= -2.6e-27) || !(y <= 1.02e-38)) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.6e-27) or not (y <= 1.02e-38):
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.6e-27) || !(y <= 1.02e-38))
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.6e-27) || ~((y <= 1.02e-38)))
		tmp = z * (-1.0 - (x / y));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.6e-27], N[Not[LessEqual[y, 1.02e-38]], $MachinePrecision]], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.60000000000000017e-27 or 1.01999999999999998e-38 < y

   1. Initial program 74.5%

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

   3. Taylor expanded in z around 0 74.5%

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

   4. Taylor expanded in z around 0 69.7%

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

      1. mul-1-neg 69.7%

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

      2. associate-/l* 78.3%

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

      3. distribute-rgt-neg-in 78.3%

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

      4. mul-1-neg 78.3%

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

      5. associate-*r/ 78.3%

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

      6. neg-mul-1 78.3%

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

      7. distribute-neg-in 78.3%

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

      8. sub-neg 78.3%

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

      9. div-sub 78.3%

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

      10. distribute-neg-frac 78.3%

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

      11. neg-mul-1 78.3%

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

      12. *-inverses 78.3%

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

      13. sub-neg 78.3%

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

      14. neg-mul-1 78.3%

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

      15. distribute-neg-in 78.3%

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

      16. +-commutative 78.3%

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

      17. distribute-neg-in 78.3%

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

      18. metadata-eval 78.3%

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

      19. unsub-neg 78.3%

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

   6. Simplified 78.3%

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

   if -2.60000000000000017e-27 < y < 1.01999999999999998e-38

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 77.9%

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

      1. +-commutative 77.9%

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

   5. Simplified 77.9%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-27} \lor \neg \left(y \leq 1.02 \cdot 10^{-38}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+101} \lor \neg \left(y \leq 8.5 \cdot 10^{+52}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.3e+101) (not (<= y 8.5e+52))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.3e+101) || !(y <= 8.5e+52)) {
		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 <= (-2.3d+101)) .or. (.not. (y <= 8.5d+52))) 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 <= -2.3e+101) || !(y <= 8.5e+52)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.3e+101) or not (y <= 8.5e+52):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.3e+101) || !(y <= 8.5e+52))
		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 <= -2.3e+101) || ~((y <= 8.5e+52)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.3e+101], N[Not[LessEqual[y, 8.5e+52]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3000000000000001e101 or 8.49999999999999994e52 < y

   1. Initial program 65.2%

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

   3. Taylor expanded in y around inf 74.7%

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

      1. neg-mul-1 74.7%

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

   5. Simplified 74.7%

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

   if -2.3000000000000001e101 < y < 8.49999999999999994e52

   1. Initial program 98.6%

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

   3. Taylor expanded in z around inf 68.9%

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

      1. +-commutative 68.9%

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

   5. Simplified 68.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+101} \lor \neg \left(y \leq 8.5 \cdot 10^{+52}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-27} \lor \neg \left(y \leq 3.1 \cdot 10^{-48}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.4e-27) (not (<= y 3.1e-48))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.4e-27) || !(y <= 3.1e-48)) {
		tmp = -z;
	} 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 ((y <= (-4.4d-27)) .or. (.not. (y <= 3.1d-48))) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.4e-27) || !(y <= 3.1e-48)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.4e-27) or not (y <= 3.1e-48):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.4e-27) || !(y <= 3.1e-48))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.4e-27) || ~((y <= 3.1e-48)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.4e-27], N[Not[LessEqual[y, 3.1e-48]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -4.39999999999999974e-27 or 3.10000000000000016e-48 < y

   1. Initial program 75.0%

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

   3. Taylor expanded in y around inf 60.0%

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

      1. neg-mul-1 60.0%

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

   5. Simplified 60.0%

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

   if -4.39999999999999974e-27 < y < 3.10000000000000016e-48

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 69.4%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-27} \lor \neg \left(y \leq 3.1 \cdot 10^{-48}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.2% 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 85.2%

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

3. Taylor expanded in y around 0 33.9%

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

Developer Target 1: 94.0% 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 2024181 
(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))))