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

Percentage Accurate: 88.8% → 99.8%

Time: 6.1s

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: 88.8% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ t_1 := \left(y + x\right) \cdot \frac{z}{z - y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-274}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-296}:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))) (t_1 (* (+ y x) (/ z (- z y)))))
   (if (<= t_0 -5e-274) t_1 (if (<= t_0 2e-296) (* (- -1.0 (/ x y)) z) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double t_1 = (y + x) * (z / (z - y));
	double tmp;
	if (t_0 <= -5e-274) {
		tmp = t_1;
	} else if (t_0 <= 2e-296) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (y + x) / (1.0d0 - (y / z))
    t_1 = (y + x) * (z / (z - y))
    if (t_0 <= (-5d-274)) then
        tmp = t_1
    else if (t_0 <= 2d-296) then
        tmp = ((-1.0d0) - (x / y)) * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double t_1 = (y + x) * (z / (z - y));
	double tmp;
	if (t_0 <= -5e-274) {
		tmp = t_1;
	} else if (t_0 <= 2e-296) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + x) / (1.0 - (y / z))
	t_1 = (y + x) * (z / (z - y))
	tmp = 0
	if t_0 <= -5e-274:
		tmp = t_1
	elif t_0 <= 2e-296:
		tmp = (-1.0 - (x / y)) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + x) / Float64(1.0 - Float64(y / z)))
	t_1 = Float64(Float64(y + x) * Float64(z / Float64(z - y)))
	tmp = 0.0
	if (t_0 <= -5e-274)
		tmp = t_1;
	elseif (t_0 <= 2e-296)
		tmp = Float64(Float64(-1.0 - Float64(x / y)) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + x) / (1.0 - (y / z));
	t_1 = (y + x) * (z / (z - y));
	tmp = 0.0;
	if (t_0 <= -5e-274)
		tmp = t_1;
	elseif (t_0 <= 2e-296)
		tmp = (-1.0 - (x / y)) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-274], t$95$1, If[LessEqual[t$95$0, 2e-296], N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5e-274 or 2e-296 < (/.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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. div-inv N/A

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

      10. neg-mul-1 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. div-inv N/A

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

      13. *-inverses N/A

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

      14. div-sub N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -5e-274 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 2e-296

   1. Initial program 23.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. div-sub N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-274}:\\ \;\;\;\;\left(y + x\right) \cdot \frac{z}{z - y}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 2 \cdot 10^{-296}:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot \frac{z}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{z - y}\\ t_1 := \left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;t\_0 \cdot y\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+69}:\\ \;\;\;\;t\_0 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ z (- z y))) (t_1 (* (- -1.0 (/ x y)) z)))
   (if (<= y -1.7e+84)
     t_1
     (if (<= y -8.5e-39) (* t_0 y) (if (<= y 1.75e+69) (* t_0 x) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = z / (z - y);
	double t_1 = (-1.0 - (x / y)) * z;
	double tmp;
	if (y <= -1.7e+84) {
		tmp = t_1;
	} else if (y <= -8.5e-39) {
		tmp = t_0 * y;
	} else if (y <= 1.75e+69) {
		tmp = t_0 * x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = z / (z - y)
    t_1 = ((-1.0d0) - (x / y)) * z
    if (y <= (-1.7d+84)) then
        tmp = t_1
    else if (y <= (-8.5d-39)) then
        tmp = t_0 * y
    else if (y <= 1.75d+69) then
        tmp = t_0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z / (z - y);
	double t_1 = (-1.0 - (x / y)) * z;
	double tmp;
	if (y <= -1.7e+84) {
		tmp = t_1;
	} else if (y <= -8.5e-39) {
		tmp = t_0 * y;
	} else if (y <= 1.75e+69) {
		tmp = t_0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z / (z - y)
	t_1 = (-1.0 - (x / y)) * z
	tmp = 0
	if y <= -1.7e+84:
		tmp = t_1
	elif y <= -8.5e-39:
		tmp = t_0 * y
	elif y <= 1.75e+69:
		tmp = t_0 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z / Float64(z - y))
	t_1 = Float64(Float64(-1.0 - Float64(x / y)) * z)
	tmp = 0.0
	if (y <= -1.7e+84)
		tmp = t_1;
	elseif (y <= -8.5e-39)
		tmp = Float64(t_0 * y);
	elseif (y <= 1.75e+69)
		tmp = Float64(t_0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z / (z - y);
	t_1 = (-1.0 - (x / y)) * z;
	tmp = 0.0;
	if (y <= -1.7e+84)
		tmp = t_1;
	elseif (y <= -8.5e-39)
		tmp = t_0 * y;
	elseif (y <= 1.75e+69)
		tmp = t_0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[y, -1.7e+84], t$95$1, If[LessEqual[y, -8.5e-39], N[(t$95$0 * y), $MachinePrecision], If[LessEqual[y, 1.75e+69], N[(t$95$0 * x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999999e84 or 1.74999999999999994e69 < y

   1. Initial program 72.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. div-sub N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 85.4

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

   5. Applied rewrites 85.4%

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

   if -1.6999999999999999e84 < y < -8.5000000000000005e-39

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. div-inv N/A

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

      10. neg-mul-1 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. div-inv N/A

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

      13. *-inverses N/A

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

      14. div-sub N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot z}{z - y}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 72.0

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

   9. Applied rewrites 72.0%

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

   if -8.5000000000000005e-39 < y < 1.74999999999999994e69

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 74.5

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

   5. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\frac{x}{\frac{z - y}{z}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.4%

      \[\leadsto \frac{z}{z - y} \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-35}:\\ \;\;\;\;\frac{z}{z - y} \cdot y\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+106}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- -1.0 (/ x y)) z)))
   (if (<= y -1.7e+84)
     t_0
     (if (<= y -6e-35)
       (* (/ z (- z y)) y)
       (if (<= y 1.45e+106) (+ y x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 - (x / y)) * z;
	double tmp;
	if (y <= -1.7e+84) {
		tmp = t_0;
	} else if (y <= -6e-35) {
		tmp = (z / (z - y)) * y;
	} else if (y <= 1.45e+106) {
		tmp = y + x;
	} 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 = ((-1.0d0) - (x / y)) * z
    if (y <= (-1.7d+84)) then
        tmp = t_0
    else if (y <= (-6d-35)) then
        tmp = (z / (z - y)) * y
    else if (y <= 1.45d+106) then
        tmp = y + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-1.0 - (x / y)) * z;
	double tmp;
	if (y <= -1.7e+84) {
		tmp = t_0;
	} else if (y <= -6e-35) {
		tmp = (z / (z - y)) * y;
	} else if (y <= 1.45e+106) {
		tmp = y + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-1.0 - (x / y)) * z
	tmp = 0
	if y <= -1.7e+84:
		tmp = t_0
	elif y <= -6e-35:
		tmp = (z / (z - y)) * y
	elif y <= 1.45e+106:
		tmp = y + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 - Float64(x / y)) * z)
	tmp = 0.0
	if (y <= -1.7e+84)
		tmp = t_0;
	elseif (y <= -6e-35)
		tmp = Float64(Float64(z / Float64(z - y)) * y);
	elseif (y <= 1.45e+106)
		tmp = Float64(y + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-1.0 - (x / y)) * z;
	tmp = 0.0;
	if (y <= -1.7e+84)
		tmp = t_0;
	elseif (y <= -6e-35)
		tmp = (z / (z - y)) * y;
	elseif (y <= 1.45e+106)
		tmp = y + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[y, -1.7e+84], t$95$0, If[LessEqual[y, -6e-35], N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.45e+106], N[(y + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999999e84 or 1.4500000000000001e106 < y

   1. Initial program 70.5%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. div-sub N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if -1.6999999999999999e84 < y < -5.99999999999999978e-35

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. div-inv N/A

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

      10. neg-mul-1 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. div-inv N/A

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

      13. *-inverses N/A

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

      14. div-sub N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot z}{z - y}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 70.9

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

   9. Applied rewrites 70.9%

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

   if -5.99999999999999978e-35 < y < 1.4500000000000001e106

   1. Initial program 98.5%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 69.5

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

   5. Applied rewrites 69.5%

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

Alternative 4: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+99}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-35}:\\ \;\;\;\;\frac{z}{z - y} \cdot y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+106}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.6e+99)
   (- z)
   (if (<= y -6e-35) (* (/ z (- z y)) y) (if (<= y 1.8e+106) (+ y x) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.6e+99) {
		tmp = -z;
	} else if (y <= -6e-35) {
		tmp = (z / (z - y)) * y;
	} else if (y <= 1.8e+106) {
		tmp = y + x;
	} 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 <= (-4.6d+99)) then
        tmp = -z
    else if (y <= (-6d-35)) then
        tmp = (z / (z - y)) * y
    else if (y <= 1.8d+106) then
        tmp = y + x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.6e+99) {
		tmp = -z;
	} else if (y <= -6e-35) {
		tmp = (z / (z - y)) * y;
	} else if (y <= 1.8e+106) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.6e+99:
		tmp = -z
	elif y <= -6e-35:
		tmp = (z / (z - y)) * y
	elif y <= 1.8e+106:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.6e+99)
		tmp = Float64(-z);
	elseif (y <= -6e-35)
		tmp = Float64(Float64(z / Float64(z - y)) * y);
	elseif (y <= 1.8e+106)
		tmp = Float64(y + x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.6e+99)
		tmp = -z;
	elseif (y <= -6e-35)
		tmp = (z / (z - y)) * y;
	elseif (y <= 1.8e+106)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.6e+99], (-z), If[LessEqual[y, -6e-35], N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.8e+106], N[(y + x), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.60000000000000038e99 or 1.8e106 < y

   1. Initial program 69.6%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if -4.60000000000000038e99 < y < -5.99999999999999978e-35

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. div-inv N/A

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

      10. neg-mul-1 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. div-inv N/A

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

      13. *-inverses N/A

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

      14. div-sub N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot z}{z - y}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 70.6

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

   9. Applied rewrites 70.6%

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

   if -5.99999999999999978e-35 < y < 1.8e106

   1. Initial program 98.5%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 69.5

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

   5. Applied rewrites 69.5%

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

Alternative 5: 68.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+21}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+106}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1e+21) (- z) (if (<= y 1.8e+106) (+ y x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+21) {
		tmp = -z;
	} else if (y <= 1.8e+106) {
		tmp = y + x;
	} 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 <= (-2.1d+21)) then
        tmp = -z
    else if (y <= 1.8d+106) then
        tmp = y + x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+21) {
		tmp = -z;
	} else if (y <= 1.8e+106) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.1e+21:
		tmp = -z
	elif y <= 1.8e+106:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.1e+21)
		tmp = Float64(-z);
	elseif (y <= 1.8e+106)
		tmp = Float64(y + x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.1e+21)
		tmp = -z;
	elseif (y <= 1.8e+106)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.1e+21], (-z), If[LessEqual[y, 1.8e+106], N[(y + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1e21 or 1.8e106 < y

   1. Initial program 72.5%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if -2.1e21 < y < 1.8e106

   1. Initial program 98.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 67.4

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

   5. Applied rewrites 67.4%

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

Alternative 6: 36.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return -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 = -z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 88.1%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

   2. lower-neg.f64 39.5

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

5. Applied rewrites 39.5%

   \[\leadsto \color{blue}{-z} \]
6. Add Preprocessing

Developer Target 1: 94.2% accurate, 0.7× 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 2024273 
(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))))