Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 92.0% → 97.4%

Time: 10.2s

Alternatives: 12

Speedup: 1.0×

• 24.8% 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: 92.0% 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: 97.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{z}{t}, y - x, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. +-commutative 94.0%

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

   2. *-commutative 94.0%

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

   3. associate-*l/ 97.5%

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

   4. fma-def 97.5%

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

3. Simplified 97.5%

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

5. Final simplification 97.5%

   \[\leadsto \mathsf{fma}\left(\frac{z}{t}, y - x, x\right) \]
6. Add Preprocessing

Alternative 2: 72.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+154} \lor \neg \left(y \leq 3.6 \cdot 10^{-75} \lor \neg \left(y \leq 5.2 \cdot 10^{+87}\right) \land y \leq 8 \cdot 10^{+181}\right):\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.3e+154)
         (not (or (<= y 3.6e-75) (and (not (<= y 5.2e+87)) (<= y 8e+181)))))
   (* (/ z t) (- y x))
   (* x (- 1.0 (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.3e+154) || !((y <= 3.6e-75) || (!(y <= 5.2e+87) && (y <= 8e+181)))) {
		tmp = (z / t) * (y - x);
	} 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 <= (-2.3d+154)) .or. (.not. (y <= 3.6d-75) .or. (.not. (y <= 5.2d+87)) .and. (y <= 8d+181))) then
        tmp = (z / t) * (y - x)
    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 <= -2.3e+154) || !((y <= 3.6e-75) || (!(y <= 5.2e+87) && (y <= 8e+181)))) {
		tmp = (z / t) * (y - x);
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -2.3e+154) or not ((y <= 3.6e-75) or (not (y <= 5.2e+87) and (y <= 8e+181))):
		tmp = (z / t) * (y - x)
	else:
		tmp = x * (1.0 - (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.3e+154) || !((y <= 3.6e-75) || (!(y <= 5.2e+87) && (y <= 8e+181))))
		tmp = Float64(Float64(z / t) * Float64(y - x));
	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 <= -2.3e+154) || ~(((y <= 3.6e-75) || (~((y <= 5.2e+87)) && (y <= 8e+181)))))
		tmp = (z / t) * (y - x);
	else
		tmp = x * (1.0 - (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.3e+154], N[Not[Or[LessEqual[y, 3.6e-75], And[N[Not[LessEqual[y, 5.2e+87]], $MachinePrecision], LessEqual[y, 8e+181]]]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3e154 or 3.6e-75 < y < 5.19999999999999997e87 or 7.9999999999999993e181 < y

   1. Initial program 90.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 69.5%

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

   4. Taylor expanded in t around 0 72.8%

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

      1. associate-/l* 70.8%

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

      2. associate-/r/ 76.0%

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

   6. Simplified 76.0%

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

   if -2.3e154 < y < 3.6e-75 or 5.19999999999999997e87 < y < 7.9999999999999993e181

   1. Initial program 95.6%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 86.1%

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

      1. mul-1-neg 86.1%

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

      2. unsub-neg 86.1%

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

   5. Simplified 86.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+154} \lor \neg \left(y \leq 3.6 \cdot 10^{-75} \lor \neg \left(y \leq 5.2 \cdot 10^{+87}\right) \land y \leq 8 \cdot 10^{+181}\right):\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{t}{y - x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* z (- y x)) t))))
   (if (<= t_1 2e+299) t_1 (/ z (/ t (- y x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + ((z * (y - x)) / t);
	double tmp;
	if (t_1 <= 2e+299) {
		tmp = t_1;
	} else {
		tmp = z / (t / (y - 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) :: t_1
    real(8) :: tmp
    t_1 = x + ((z * (y - x)) / t)
    if (t_1 <= 2d+299) then
        tmp = t_1
    else
        tmp = z / (t / (y - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + ((z * (y - x)) / t);
	double tmp;
	if (t_1 <= 2e+299) {
		tmp = t_1;
	} else {
		tmp = z / (t / (y - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + ((z * (y - x)) / t)
	tmp = 0
	if t_1 <= 2e+299:
		tmp = t_1
	else:
		tmp = z / (t / (y - x))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(z * Float64(y - x)) / t))
	tmp = 0.0
	if (t_1 <= 2e+299)
		tmp = t_1;
	else
		tmp = Float64(z / Float64(t / Float64(y - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + ((z * (y - x)) / t);
	tmp = 0.0;
	if (t_1 <= 2e+299)
		tmp = t_1;
	else
		tmp = z / (t / (y - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+299], t$95$1, N[(z / N[(t / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 2.0000000000000001e299

   1. Initial program 98.5%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   if 2.0000000000000001e299 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

   1. Initial program 74.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 89.9%

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

   4. Taylor expanded in t around 0 74.9%

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

      1. associate-/l* 92.0%

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

   6. Simplified 92.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{z \cdot \left(y - x\right)}{t} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;x + \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{t}{y - x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-136} \lor \neg \left(x \leq 7.2 \cdot 10^{-201}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.7e-136) (not (<= x 7.2e-201)))
   (* x (- 1.0 (/ z t)))
   (* z (/ y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.7e-136) || !(x <= 7.2e-201)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = z * (y / 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 ((x <= (-2.7d-136)) .or. (.not. (x <= 7.2d-201))) then
        tmp = x * (1.0d0 - (z / t))
    else
        tmp = z * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.7e-136) || !(x <= 7.2e-201)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = z * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.7e-136) or not (x <= 7.2e-201):
		tmp = x * (1.0 - (z / t))
	else:
		tmp = z * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.7e-136) || !(x <= 7.2e-201))
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	else
		tmp = Float64(z * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.7e-136) || ~((x <= 7.2e-201)))
		tmp = x * (1.0 - (z / t));
	else
		tmp = z * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.7e-136], N[Not[LessEqual[x, 7.2e-201]], $MachinePrecision]], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6999999999999998e-136 or 7.20000000000000063e-201 < x

   1. Initial program 93.2%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 81.5%

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

      1. mul-1-neg 81.5%

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

      2. unsub-neg 81.5%

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

   5. Simplified 81.5%

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

   if -2.6999999999999998e-136 < x < 7.20000000000000063e-201

   1. Initial program 96.7%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 69.4%

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

   4. Taylor expanded in y around inf 67.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-136} \lor \neg \left(x \leq 7.2 \cdot 10^{-201}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.62 \cdot 10^{+66} \lor \neg \left(x \leq 4600000000000\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.62e+66) (not (<= x 4600000000000.0)))
   (* x (- 1.0 (/ z t)))
   (+ x (* z (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.62e+66) || !(x <= 4600000000000.0)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + (z * (y / 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 ((x <= (-1.62d+66)) .or. (.not. (x <= 4600000000000.0d0))) then
        tmp = x * (1.0d0 - (z / t))
    else
        tmp = x + (z * (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.62e+66) || !(x <= 4600000000000.0)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + (z * (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.62e+66) or not (x <= 4600000000000.0):
		tmp = x * (1.0 - (z / t))
	else:
		tmp = x + (z * (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.62e+66) || !(x <= 4600000000000.0))
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	else
		tmp = Float64(x + Float64(z * Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.62e+66) || ~((x <= 4600000000000.0)))
		tmp = x * (1.0 - (z / t));
	else
		tmp = x + (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.62e+66], N[Not[LessEqual[x, 4600000000000.0]], $MachinePrecision]], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.62e66 or 4.6e12 < x

   1. Initial program 90.1%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 93.9%

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

      1. mul-1-neg 93.9%

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

      2. unsub-neg 93.9%

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

   5. Simplified 93.9%

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

   if -1.62e66 < x < 4.6e12

   1. Initial program 97.2%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 85.1%

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

      1. associate-/l* 85.6%

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

      2. associate-/r/ 83.8%

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

   5. Simplified 83.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.62 \cdot 10^{+66} \lor \neg \left(x \leq 4600000000000\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+69} \lor \neg \left(x \leq 680000000000\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.6e+69) (not (<= x 680000000000.0)))
   (* x (- 1.0 (/ z t)))
   (+ x (* (/ z t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.6e+69) || !(x <= 680000000000.0)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + ((z / t) * y);
	}
	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 <= (-4.6d+69)) .or. (.not. (x <= 680000000000.0d0))) then
        tmp = x * (1.0d0 - (z / t))
    else
        tmp = x + ((z / t) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.6e+69) || !(x <= 680000000000.0)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + ((z / t) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.6e+69) or not (x <= 680000000000.0):
		tmp = x * (1.0 - (z / t))
	else:
		tmp = x + ((z / t) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.6e+69) || !(x <= 680000000000.0))
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	else
		tmp = Float64(x + Float64(Float64(z / t) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4.6e+69) || ~((x <= 680000000000.0)))
		tmp = x * (1.0 - (z / t));
	else
		tmp = x + ((z / t) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.6e+69], N[Not[LessEqual[x, 680000000000.0]], $MachinePrecision]], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.60000000000000033e69 or 6.8e11 < x

   1. Initial program 90.1%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 93.9%

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

      1. mul-1-neg 93.9%

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

      2. unsub-neg 93.9%

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

   5. Simplified 93.9%

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

   if -4.60000000000000033e69 < x < 6.8e11

   1. Initial program 97.2%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 85.1%

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

      1. *-commutative 85.1%

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

   5. Simplified 85.1%

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

      1. associate-/l* 83.7%

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

      2. associate-/r/ 85.6%

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

   7. Applied egg-rr 85.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+69} \lor \neg \left(x \leq 680000000000\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+60} \lor \neg \left(z \leq 2.6 \cdot 10^{+104}\right):\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.3e+60) (not (<= z 2.6e+104))) (* z (/ y t)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.3e+60) || !(z <= 2.6e+104)) {
		tmp = z * (y / 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 ((z <= (-1.3d+60)) .or. (.not. (z <= 2.6d+104))) then
        tmp = z * (y / 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 ((z <= -1.3e+60) || !(z <= 2.6e+104)) {
		tmp = z * (y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.3e+60) or not (z <= 2.6e+104):
		tmp = z * (y / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.3e+60) || !(z <= 2.6e+104))
		tmp = Float64(z * Float64(y / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.3e+60) || ~((z <= 2.6e+104)))
		tmp = z * (y / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.3e+60], N[Not[LessEqual[z, 2.6e+104]], $MachinePrecision]], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.30000000000000004e60 or 2.6e104 < z

   1. Initial program 85.8%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 91.9%

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

   4. Taylor expanded in y around inf 57.4%

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

   if -1.30000000000000004e60 < z < 2.6e104

   1. Initial program 98.2%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 59.8%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+60} \lor \neg \left(z \leq 2.6 \cdot 10^{+104}\right):\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+130}:\\ \;\;\;\;z \cdot \frac{-x}{t}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+104}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.9e+130) (* z (/ (- x) t)) (if (<= z 3.8e+104) x (* z (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.9e+130) {
		tmp = z * (-x / t);
	} else if (z <= 3.8e+104) {
		tmp = x;
	} else {
		tmp = z * (y / 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 (z <= (-1.9d+130)) then
        tmp = z * (-x / t)
    else if (z <= 3.8d+104) then
        tmp = x
    else
        tmp = z * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.9e+130) {
		tmp = z * (-x / t);
	} else if (z <= 3.8e+104) {
		tmp = x;
	} else {
		tmp = z * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.9e+130:
		tmp = z * (-x / t)
	elif z <= 3.8e+104:
		tmp = x
	else:
		tmp = z * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.9e+130)
		tmp = Float64(z * Float64(Float64(-x) / t));
	elseif (z <= 3.8e+104)
		tmp = x;
	else
		tmp = Float64(z * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.9e+130)
		tmp = z * (-x / t);
	elseif (z <= 3.8e+104)
		tmp = x;
	else
		tmp = z * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.9e+130], N[(z * N[((-x) / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e+104], x, N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9000000000000001e130

   1. Initial program 88.5%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 56.8%

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

      1. mul-1-neg 56.8%

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

      2. distribute-lft-neg-out 56.8%

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

      3. *-commutative 56.8%

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

   5. Simplified 56.8%

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

      1. div-inv 56.8%

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

      2. *-commutative 56.8%

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

      3. add-sqr-sqrt 31.3%

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

      4. sqrt-unprod 34.4%

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

      5. sqr-neg 34.4%

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

      6. sqrt-unprod 3.2%

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

      7. add-sqr-sqrt 5.9%

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

      8. *-commutative 5.9%

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

      9. remove-double-neg 5.9%

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

      10. distribute-rgt-neg-out 5.9%

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

      11. cancel-sign-sub-inv 5.9%

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

      12. *-commutative 5.9%

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

      13. associate-*l* 10.7%

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

      14. div-inv 10.7%

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

      15. add-sqr-sqrt 2.7%

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

      16. sqrt-unprod 27.8%

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

      17. sqr-neg 27.8%

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

      18. sqrt-unprod 32.6%

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

      19. add-sqr-sqrt 68.4%

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

   7. Applied egg-rr 68.4%

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

   8. Taylor expanded in z around inf 56.8%

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

      1. mul-1-neg 56.8%

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

      2. associate-*l/ 63.7%

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

      3. distribute-rgt-neg-in 63.7%

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

   10. Simplified 63.7%

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

   if -1.9000000000000001e130 < z < 3.79999999999999969e104

   1. Initial program 97.8%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 58.6%

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

   if 3.79999999999999969e104 < z

   1. Initial program 82.1%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 91.9%

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

   4. Taylor expanded in y around inf 57.3%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+130}:\\ \;\;\;\;z \cdot \frac{-x}{t}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+104}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \frac{z}{-t}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+104}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.15e+89) (* x (/ z (- t))) (if (<= z 2.4e+104) x (* z (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.15e+89) {
		tmp = x * (z / -t);
	} else if (z <= 2.4e+104) {
		tmp = x;
	} else {
		tmp = z * (y / 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 (z <= (-1.15d+89)) then
        tmp = x * (z / -t)
    else if (z <= 2.4d+104) then
        tmp = x
    else
        tmp = z * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.15e+89) {
		tmp = x * (z / -t);
	} else if (z <= 2.4e+104) {
		tmp = x;
	} else {
		tmp = z * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.15e+89:
		tmp = x * (z / -t)
	elif z <= 2.4e+104:
		tmp = x
	else:
		tmp = z * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.15e+89)
		tmp = Float64(x * Float64(z / Float64(-t)));
	elseif (z <= 2.4e+104)
		tmp = x;
	else
		tmp = Float64(z * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.15e+89)
		tmp = x * (z / -t);
	elseif (z <= 2.4e+104)
		tmp = x;
	else
		tmp = z * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.15e+89], N[(x * N[(z / (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.4e+104], x, N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1499999999999999e89

   1. Initial program 87.2%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 54.1%

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

      1. mul-1-neg 54.1%

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

      2. distribute-lft-neg-out 54.1%

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

      3. *-commutative 54.1%

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

   5. Simplified 54.1%

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

      1. div-inv 54.0%

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

      2. *-commutative 54.0%

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

      3. add-sqr-sqrt 28.5%

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

      4. sqrt-unprod 31.4%

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

      5. sqr-neg 31.4%

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

      6. sqrt-unprod 5.2%

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

      7. add-sqr-sqrt 7.8%

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

      8. *-commutative 7.8%

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

      9. remove-double-neg 7.8%

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

      10. distribute-rgt-neg-out 7.8%

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

      11. cancel-sign-sub-inv 7.8%

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

      12. *-commutative 7.8%

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

      13. associate-*l* 14.3%

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

      14. div-inv 14.3%

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

      15. add-sqr-sqrt 4.8%

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

      16. sqrt-unprod 25.4%

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

      17. sqr-neg 25.4%

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

      18. sqrt-unprod 31.9%

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

      19. add-sqr-sqrt 66.7%

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

   7. Applied egg-rr 66.7%

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

   8. Taylor expanded in z around inf 51.9%

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

      1. mul-1-neg 51.9%

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

      2. associate-*l/ 58.1%

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

      3. distribute-rgt-neg-in 58.1%

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

   10. Simplified 58.1%

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

       1. add-sqr-sqrt 58.1%

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

       2. sqrt-unprod 56.0%

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

       3. sqr-neg 56.0%

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

       4. sqrt-unprod 0.0%

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

       5. add-sqr-sqrt 5.6%

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

       6. associate-*l/ 5.6%

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

       7. associate-*r/ 12.1%

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

       8. frac-2neg 12.1%

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

       9. associate-*r/ 5.6%

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

       10. *-commutative 5.6%

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

       11. add-sqr-sqrt 5.6%

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

       12. sqrt-unprod 11.8%

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

       13. sqr-neg 11.8%

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

       14. sqrt-unprod 0.0%

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

       15. add-sqr-sqrt 51.9%

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

   12. Applied egg-rr 51.9%

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

       1. associate-/l* 58.2%

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

       2. associate-/r/ 64.6%

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

   14. Simplified 64.6%

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

   if -1.1499999999999999e89 < z < 2.4e104

   1. Initial program 98.3%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 59.3%

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

   if 2.4e104 < z

   1. Initial program 82.1%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 91.9%

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

   4. Taylor expanded in y around inf 57.3%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \frac{z}{-t}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+104}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.4% 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 94.0%

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

   1. associate-/l* 97.5%

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

3. Simplified 97.5%

   \[\leadsto \color{blue}{x + \frac{y - x}{\frac{t}{z}}} \]
4. Add Preprocessing

5. Final simplification 97.5%

   \[\leadsto x + \frac{y - x}{\frac{t}{z}} \]
6. Add Preprocessing

Alternative 11: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. +-commutative 94.0%

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

   2. *-commutative 94.0%

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

   3. associate-*l/ 97.5%

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

   4. fma-def 97.5%

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

3. Simplified 97.5%

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

   1. fma-udef 97.5%

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

   2. *-commutative 97.5%

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

6. Applied egg-rr 97.5%

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

7. Final simplification 97.5%

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

Alternative 12: 38.9% 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 94.0%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing

3. Taylor expanded in z around 0 42.6%

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

4. Final simplification 42.6%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 97.6% accurate, 0.5× 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 2024019 
(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)))