Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 92.4% → 95.6%

Time: 6.8s

Alternatives: 10

Speedup: 1.0×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) z) t))))
   (if (<= t_1 (- INFINITY)) (+ x (* z (/ (- y x) t))) t_1)))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (((y - x) * z) / t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x + (z * ((y - x) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + (((y - x) * z) / t);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = x + (z * ((y - x) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (((y - x) * z) / t)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = x + (z * ((y - x) / t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * z) / t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + (((y - x) * z) / t);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = x + (z * ((y - x) / t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -inf.0

   1. Initial program 71.7%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

   1. Initial program 97.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq -\infty:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array} \]

Alternative 2: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-108}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{x}{\frac{-t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1e-108)
   (+ x (* y (/ z t)))
   (if (<= y 2.4e-48) (+ x (/ x (/ (- t) z))) (+ x (/ y (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1e-108) {
		tmp = x + (y * (z / t));
	} else if (y <= 2.4e-48) {
		tmp = x + (x / (-t / z));
	} else {
		tmp = x + (y / (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1d-108)) then
        tmp = x + (y * (z / t))
    else if (y <= 2.4d-48) then
        tmp = x + (x / (-t / z))
    else
        tmp = x + (y / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1e-108) {
		tmp = x + (y * (z / t));
	} else if (y <= 2.4e-48) {
		tmp = x + (x / (-t / z));
	} else {
		tmp = x + (y / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1e-108:
		tmp = x + (y * (z / t))
	elif y <= 2.4e-48:
		tmp = x + (x / (-t / z))
	else:
		tmp = x + (y / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1e-108)
		tmp = Float64(x + Float64(y * Float64(z / t)));
	elseif (y <= 2.4e-48)
		tmp = Float64(x + Float64(x / Float64(Float64(-t) / z)));
	else
		tmp = Float64(x + Float64(y / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1e-108)
		tmp = x + (y * (z / t));
	elseif (y <= 2.4e-48)
		tmp = x + (x / (-t / z));
	else
		tmp = x + (y / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1e-108], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e-48], N[(x + N[(x / N[((-t) / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.00000000000000004e-108

   1. Initial program 92.6%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Step-by-step derivation

      1. associate-*l/ 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in y around inf 87.8%

      \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]
   5. Step-by-step derivation

      1. associate-*r/ 92.5%

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

   6. Simplified 92.5%

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

   if -1.00000000000000004e-108 < y < 2.4e-48

   1. Initial program 94.8%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Step-by-step derivation

      1. associate-*l/ 93.6%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in y around 0 83.2%

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

      1. neg-mul-1 83.2%

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

      2. distribute-neg-frac 83.2%

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

   6. Simplified 83.2%

      \[\leadsto x + \color{blue}{\frac{-x}{t}} \cdot z \]
   7. Step-by-step derivation

      1. *-commutative 83.2%

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

      2. frac-2neg 83.2%

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

      3. remove-double-neg 83.2%

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

      4. associate-*r/ 83.9%

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

   8. Applied egg-rr 83.9%

      \[\leadsto x + \color{blue}{\frac{z \cdot x}{-t}} \]
   9. Step-by-step derivation

      1. *-commutative 83.9%

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

      2. associate-/l* 88.6%

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

      3. distribute-frac-neg 88.6%

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

   10. Simplified 88.6%

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

   if 2.4e-48 < y

   1. Initial program 92.7%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Step-by-step derivation

      1. associate-*l/ 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in y around inf 87.3%

      \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]
   5. Step-by-step derivation

      1. associate-*r/ 92.0%

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

   6. Simplified 92.0%

      \[\leadsto x + \color{blue}{y \cdot \frac{z}{t}} \]
   7. Step-by-step derivation

      1. clear-num 92.0%

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

      2. un-div-inv 92.1%

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

   8. Applied egg-rr 92.1%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-108}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{x}{\frac{-t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 3: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-108} \lor \neg \left(y \leq 4.3 \cdot 10^{-49}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -4e-108) (not (<= y 4.3e-49)))
   (+ x (* y (/ z t)))
   (* x (- 1.0 (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4e-108) || !(y <= 4.3e-49)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-4d-108)) .or. (.not. (y <= 4.3d-49))) then
        tmp = x + (y * (z / t))
    else
        tmp = x * (1.0d0 - (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4e-108) || !(y <= 4.3e-49)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -4e-108) or not (y <= 4.3e-49):
		tmp = x + (y * (z / t))
	else:
		tmp = x * (1.0 - (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4e-108) || !(y <= 4.3e-49))
		tmp = Float64(x + Float64(y * Float64(z / t)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -4e-108) || ~((y <= 4.3e-49)))
		tmp = x + (y * (z / t));
	else
		tmp = x * (1.0 - (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -4e-108], N[Not[LessEqual[y, 4.3e-49]], $MachinePrecision]], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.00000000000000016e-108 or 4.30000000000000016e-49 < y

   1. Initial program 92.6%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Step-by-step derivation

      1. associate-*l/ 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in y around inf 87.5%

      \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]
   5. Step-by-step derivation

      1. associate-*r/ 92.3%

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

   6. Simplified 92.3%

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

   if -4.00000000000000016e-108 < y < 4.30000000000000016e-49

   1. Initial program 94.8%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Step-by-step derivation

      1. +-commutative 94.8%

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

      2. remove-double-neg 94.8%

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

      3. unsub-neg 94.8%

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

      4. associate-*r/ 97.1%

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

      5. fma-neg 97.1%

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

      6. remove-double-neg 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in y around 0 83.9%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x \cdot z}{t}} \]
   5. Step-by-step derivation

      1. mul-1-neg 83.9%

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

      2. associate-*r/ 88.6%

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

      3. *-rgt-identity 88.6%

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

      4. distribute-rgt-neg-in 88.6%

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

      5. mul-1-neg 88.6%

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

      6. distribute-lft-in 88.6%

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

      7. mul-1-neg 88.6%

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

      8. unsub-neg 88.6%

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

   6. Simplified 88.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-108} \lor \neg \left(y \leq 4.3 \cdot 10^{-49}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \]

Alternative 4: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-107}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{-48}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.32e-107)
   (+ x (* y (/ z t)))
   (if (<= y 1.26e-48) (* x (- 1.0 (/ z t))) (+ x (/ y (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.32e-107) {
		tmp = x + (y * (z / t));
	} else if (y <= 1.26e-48) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + (y / (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.32d-107)) then
        tmp = x + (y * (z / t))
    else if (y <= 1.26d-48) then
        tmp = x * (1.0d0 - (z / t))
    else
        tmp = x + (y / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.32e-107) {
		tmp = x + (y * (z / t));
	} else if (y <= 1.26e-48) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + (y / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.32e-107:
		tmp = x + (y * (z / t))
	elif y <= 1.26e-48:
		tmp = x * (1.0 - (z / t))
	else:
		tmp = x + (y / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.32e-107)
		tmp = Float64(x + Float64(y * Float64(z / t)));
	elseif (y <= 1.26e-48)
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	else
		tmp = Float64(x + Float64(y / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.32e-107)
		tmp = x + (y * (z / t));
	elseif (y <= 1.26e-48)
		tmp = x * (1.0 - (z / t));
	else
		tmp = x + (y / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.32e-107], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.26e-48], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3200000000000001e-107

   1. Initial program 92.6%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Step-by-step derivation

      1. associate-*l/ 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in y around inf 87.8%

      \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]
   5. Step-by-step derivation

      1. associate-*r/ 92.5%

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

   6. Simplified 92.5%

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

   if -1.3200000000000001e-107 < y < 1.2599999999999999e-48

   1. Initial program 94.8%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Step-by-step derivation

      1. +-commutative 94.8%

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

      2. remove-double-neg 94.8%

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

      3. unsub-neg 94.8%

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

      4. associate-*r/ 97.1%

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

      5. fma-neg 97.1%

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

      6. remove-double-neg 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in y around 0 83.9%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x \cdot z}{t}} \]
   5. Step-by-step derivation

      1. mul-1-neg 83.9%

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

      2. associate-*r/ 88.6%

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

      3. *-rgt-identity 88.6%

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

      4. distribute-rgt-neg-in 88.6%

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

      5. mul-1-neg 88.6%

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

      6. distribute-lft-in 88.6%

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

      7. mul-1-neg 88.6%

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

      8. unsub-neg 88.6%

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

   6. Simplified 88.6%

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

   if 1.2599999999999999e-48 < y

   1. Initial program 92.7%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Step-by-step derivation

      1. associate-*l/ 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in y around inf 87.3%

      \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]
   5. Step-by-step derivation

      1. associate-*r/ 92.0%

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

   6. Simplified 92.0%

      \[\leadsto x + \color{blue}{y \cdot \frac{z}{t}} \]
   7. Step-by-step derivation

      1. clear-num 92.0%

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

      2. un-div-inv 92.1%

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

   8. Applied egg-rr 92.1%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-107}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{-48}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 5: 50.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-32}:\\ \;\;\;\;z \cdot \frac{-x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.8e-44) x (if (<= t 2.35e-32) (* z (/ (- x) t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.8e-44) {
		tmp = x;
	} else if (t <= 2.35e-32) {
		tmp = z * (-x / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3.8d-44)) then
        tmp = x
    else if (t <= 2.35d-32) then
        tmp = z * (-x / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.8e-44) {
		tmp = x;
	} else if (t <= 2.35e-32) {
		tmp = z * (-x / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.8e-44:
		tmp = x
	elif t <= 2.35e-32:
		tmp = z * (-x / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.8e-44)
		tmp = x;
	elseif (t <= 2.35e-32)
		tmp = Float64(z * Float64(Float64(-x) / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3.8e-44)
		tmp = x;
	elseif (t <= 2.35e-32)
		tmp = z * (-x / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.8e-44], x, If[LessEqual[t, 2.35e-32], N[(z * N[((-x) / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -3.8000000000000001e-44 or 2.3500000000000001e-32 < t

   1. Initial program 89.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Step-by-step derivation

      1. +-commutative 89.9%

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

      2. remove-double-neg 89.9%

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

      3. unsub-neg 89.9%

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

      4. associate-*r/ 98.5%

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

      5. fma-neg 98.5%

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

      6. remove-double-neg 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in z around 0 63.6%

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

   if -3.8000000000000001e-44 < t < 2.3500000000000001e-32

   1. Initial program 98.1%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Step-by-step derivation

      1. +-commutative 98.1%

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

      2. remove-double-neg 98.1%

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

      3. unsub-neg 98.1%

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

      4. associate-*r/ 97.3%

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

      5. fma-neg 97.4%

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

      6. remove-double-neg 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in y around 0 46.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x \cdot z}{t}} \]
   5. Step-by-step derivation

      1. mul-1-neg 46.6%

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

      2. associate-*r/ 50.7%

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

      3. *-rgt-identity 50.7%

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

      4. distribute-rgt-neg-in 50.7%

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

      5. mul-1-neg 50.7%

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

      6. distribute-lft-in 50.7%

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

      7. mul-1-neg 50.7%

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

      8. unsub-neg 50.7%

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

   6. Simplified 50.7%

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

   7. Taylor expanded in z around inf 39.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{t}} \]
   8. Step-by-step derivation

      1. mul-1-neg 39.0%

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

      2. *-commutative 39.0%

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

      3. associate-*r/ 36.6%

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

      4. distribute-rgt-neg-in 36.6%

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

      5. distribute-neg-frac 36.6%

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

   9. Simplified 36.6%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-32}:\\ \;\;\;\;z \cdot \frac{-x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 51.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.05e-43) x (if (<= t 1.55e-33) (* x (/ (- z) t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.05e-43) {
		tmp = x;
	} else if (t <= 1.55e-33) {
		tmp = x * (-z / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.05d-43)) then
        tmp = x
    else if (t <= 1.55d-33) then
        tmp = x * (-z / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.05e-43) {
		tmp = x;
	} else if (t <= 1.55e-33) {
		tmp = x * (-z / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.05e-43:
		tmp = x
	elif t <= 1.55e-33:
		tmp = x * (-z / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.05e-43)
		tmp = x;
	elseif (t <= 1.55e-33)
		tmp = Float64(x * Float64(Float64(-z) / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.05e-43)
		tmp = x;
	elseif (t <= 1.55e-33)
		tmp = x * (-z / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.05e-43], x, If[LessEqual[t, 1.55e-33], N[(x * N[((-z) / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -1.05e-43 or 1.54999999999999998e-33 < t

   1. Initial program 89.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Step-by-step derivation

      1. +-commutative 89.9%

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

      2. remove-double-neg 89.9%

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

      3. unsub-neg 89.9%

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

      4. associate-*r/ 98.5%

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

      5. fma-neg 98.5%

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

      6. remove-double-neg 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in z around 0 63.6%

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

   if -1.05e-43 < t < 1.54999999999999998e-33

   1. Initial program 98.1%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Step-by-step derivation

      1. +-commutative 98.1%

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

      2. remove-double-neg 98.1%

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

      3. unsub-neg 98.1%

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

      4. associate-*r/ 97.3%

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

      5. fma-neg 97.4%

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

      6. remove-double-neg 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in y around 0 46.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x \cdot z}{t}} \]
   5. Step-by-step derivation

      1. mul-1-neg 46.6%

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

      2. associate-*r/ 50.7%

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

      3. *-rgt-identity 50.7%

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

      4. distribute-rgt-neg-in 50.7%

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

      5. mul-1-neg 50.7%

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

      6. distribute-lft-in 50.7%

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

      7. mul-1-neg 50.7%

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

      8. unsub-neg 50.7%

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

   6. Simplified 50.7%

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

   7. Taylor expanded in z around inf 39.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{t}} \]
   8. Step-by-step derivation

      1. associate-*r/ 39.0%

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

      2. *-commutative 39.0%

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

      3. neg-mul-1 39.0%

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

      4. distribute-neg-frac 39.0%

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

      5. associate-/l* 36.6%

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

      6. associate-/r/ 43.2%

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

      7. distribute-rgt-neg-in 43.2%

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

   9. Simplified 43.2%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 93.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.5%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Step-by-step derivation

   1. associate-*l/ 93.7%

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

3. Simplified 93.7%

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

4. Final simplification 93.7%

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

Alternative 8: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.5%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Step-by-step derivation

   1. associate-/l* 98.2%

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

3. Simplified 98.2%

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

4. Final simplification 98.2%

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

Alternative 9: 65.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - \frac{z}{t}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.5%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Step-by-step derivation

   1. +-commutative 93.5%

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

   2. remove-double-neg 93.5%

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

   3. unsub-neg 93.5%

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

   4. associate-*r/ 98.0%

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

   5. fma-neg 98.0%

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

   6. remove-double-neg 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in y around 0 59.1%

   \[\leadsto \color{blue}{x + -1 \cdot \frac{x \cdot z}{t}} \]
5. Step-by-step derivation

   1. mul-1-neg 59.1%

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

   2. associate-*r/ 62.7%

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

   3. *-rgt-identity 62.7%

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

   4. distribute-rgt-neg-in 62.7%

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

   5. mul-1-neg 62.7%

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

   6. distribute-lft-in 62.8%

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

   7. mul-1-neg 62.8%

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

   8. unsub-neg 62.8%

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

6. Simplified 62.8%

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

7. Final simplification 62.8%

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

Alternative 10: 38.3% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.5%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Step-by-step derivation

   1. +-commutative 93.5%

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

   2. remove-double-neg 93.5%

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

   3. unsub-neg 93.5%

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

   4. associate-*r/ 98.0%

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

   5. fma-neg 98.0%

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

   6. remove-double-neg 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in z around 0 39.9%

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

5. Final simplification 39.9%

   \[\leadsto x \]

Developer target: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< x -9.025511195533005e-135)
   (- x (* (/ z t) (- x y)))
   (if (< x 4.275032163700715e-250)
     (+ x (* (/ (- y x) t) z))
     (+ x (/ (- y x) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x < (-9.025511195533005d-135)) then
        tmp = x - ((z / t) * (x - y))
    else if (x < 4.275032163700715d-250) then
        tmp = x + (((y - x) / t) * z)
    else
        tmp = x + ((y - x) / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x < -9.025511195533005e-135:
		tmp = x - ((z / t) * (x - y))
	elif x < 4.275032163700715e-250:
		tmp = x + (((y - x) / t) * z)
	else:
		tmp = x + ((y - x) / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x < -9.025511195533005e-135)
		tmp = Float64(x - Float64(Float64(z / t) * Float64(x - y)));
	elseif (x < 4.275032163700715e-250)
		tmp = Float64(x + Float64(Float64(Float64(y - x) / t) * z));
	else
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x < -9.025511195533005e-135)
		tmp = x - ((z / t) * (x - y));
	elseif (x < 4.275032163700715e-250)
		tmp = x + (((y - x) / t) * z);
	else
		tmp = x + ((y - x) / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[x, -9.025511195533005e-135], N[(x - N[(N[(z / t), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[x, 4.275032163700715e-250], N[(x + N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023336 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

  (+ x (/ (* (- y x) z) t)))